Holomorphic Operator Functions of One Variable and Applications: Methods from Complex Analysis in Several Variables (Operator Theory: Advances and Applications)

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Author: Israel Gohberg | Jürgen Leiterer

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Operator Theory: Advances and Applications Vol. 192

Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv Israel

Editorial Board: D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France)

Honorary and Advisory Editorial Board: L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

Holomorphic Operator Functions of One Variable and Applications Methods from Complex Analysis in Several Variables

Israel Gohberg Jürgen Leiterer

Birkhäuser Basel · Boston · Berlin

Authors: Israel Gohberg Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 69978 Ramat Aviv Israel e-mail: [email protected]

Jürgen Leiterer Institut für Mathematik Humboldt-Universität zu Berlin Rudower Chaussee 25 12489 Berlin Germany e-mail: [email protected]

2000 Mathematical Subject Classification: 47-02, 47A56, 47A68, 47B35; 30-02, 30E05; 32L05, 32L10, 32L20

Library of Congress Control Number: 2009927556

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-0125-2 Birkhäuser Verlag AG, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany

ISBN 978-3-0346-0125-2

e-ISBN 978-3-0346-0126-9

987654321

www.birkhauser.ch

Dedicated to the memory of our friend, student and colleague

Georg Heinig

Contents Preface

xi

Introduction

xiii

Notation

xix

1

2

Elementary properties of holomorphic functions 1.1 Definition and first properties . . . . . . . . . . . . . 1.2 The maximum principle . . . . . . . . . . . . . . . . 1.3 Contour integrals . . . . . . . . . . . . . . . . . . . . 1.4 The Cauchy integral theorem . . . . . . . . . . . . . 1.5 The Cauchy formula . . . . . . . . . . . . . . . . . . 1.6 The Hahn-Banach criterion . . . . . . . . . . . . . . 1.7 A criterion for the holomorphy of operator functions 1.8 Power series . . . . . . . . . . . . . . . . . . . . . . . 1.9 Laurent series . . . . . . . . . . . . . . . . . . . . . . 1.10 Isolated singularities . . . . . . . . . . . . . . . . . . 1.11 Comments . . . . . . . . . . . . . . . . . . . . . . . .

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Solution of ∂u = f and applications 2.1 The Pompeiju formula for solutions of ∂u = f on compact sets . . 2.2 Runge approximation . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Solution of ∂u = f on open sets . . . . . . . . . . . . . . . . . . . . 2.4 OE -cocycles and the Mittag-Leffler theorem . . . . . . . . . . . . . 2.5 Runge approximation for invertible scalar functions and the Weierstrass product theorem . . . . . . . . . . . . . . . . . . . . . 2.6 OE -cocycles with prescribed zeros and a stronger version of the Mittag-Leffler theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Generalization of the Weierstrass product theorem . . . . . . . . . 2.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 8 11 12 15 18 19 23 25 28 29 29 37 41 43 44 52 54 58

viii 3

Contents Splitting and factorization with respect to a contour 3.1 Splitting with respect to a contour . . . . . . . . 3.2 Splitting and the Cauchy Integral . . . . . . . . . 3.3 H¨ older continuous functions split . . . . . . . . . 3.4 The splitting behavior of differentiable functions 3.5 Approximation of H¨ older continuous functions . . 3.6 Example: A non-splitting continuous function . . 3.7 The additive local principle . . . . . . . . . . . . 3.8 Factorization of scalar functions with respect to a First remarks . . . . . . . . . . . . . . . . . . . . 3.9 Factorization of H¨ older functions . . . . . . . . . 3.10 Factorization of Wiener functions . . . . . . . . . 3.11 The multiplicative local principle . . . . . . . . . 3.12 Comments . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . contour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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59 59 61 65 72 75 77 82

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84 89 91 93 95

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97 97 101 106 111 112

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113 114 120 123 125 129 131 133 137

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141 143 147 151 153 156 157

Families of subspaces 6.1 The gap metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Kernel and image of operator functions . . . . . . . . . . . . . . . 6.3 Holomorphic sections of continuous families of subspaces . . . . . .

159 159 172 184

4 The 4.1 4.2 4.3 4.4 4.5

Rouché theorem for operator functions Finite meromorphic Fredholm functions . . . . . Invertible finite meromorphic Fredholm functions Smith factorization . . . . . . . . . . . . . . . . . The Rouché theorem . . . . . . . . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

5 Multiplicative cocycles (OG -cocycles) 5.1 Topological properties of GL(E) . . . . . . . . . . . . . . . . . . 5.2 Two factorization lemmas . . . . . . . . . . . . . . . . . . . . . . 5.3 OE -cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Runge approximation of G-valued functions. First steps . . . . . 5.5 The Cartan lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 OG -cocycles. Definitions and statement of the main result . . . . 5.7 Refinement of the covering . . . . . . . . . . . . . . . . . . . . . . 5.8 Exhausting by compact sets . . . . . . . . . . . . . . . . . . . . . 5.9 Proof of the main theorem in the case of simply connected open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Runge approximation of G-valued functions. General case . . . . 5.11 Proof of the main theorem in the general case . . . . . . . . . . . ∞ 5.12 OG (E) -cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Weierstrass theorems . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Weierstrass theorems for G ∞ (E) and G ω (E)-valued functions . . 5.15 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Contents

ix

6.4 6.5

Holomorphic families of subspaces . . . . . . . . . . . . . . . . . . Example: A holomorphic family of subspaces with jumping isomorphism type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Injective families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Shubin families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Complemented families . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Finite dimensional and finite codimensional families . . . . . . . . 6.10 One-sided and generalized invertible holomorphic operator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Example: A globally non-trivial complemented holomorphic family of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8

185 201 203 204 206 209 211 214 216

Plemelj-Muschelishvili factorization 7.1 Definitions and first remarks about factorization . . . . . . . . . . 7.2 The algebra of Wiener functions and other splitting R-algebras . . 7.3 H¨ older continuous and differentiable functions . . . . . . . . . . . . 7.4 Reduction of the factorization problem to functions, holomorphic and invertible on C∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Factorization of holomorphic functions close to the unit . . . . . . 7.6 Reduction of the factorization problem to polynomials in z and 1/z 7.7 The finite dimensional case . . . . . . . . . . . . . . . . . . . . . . 7.8 Factorization of G ∞ (E)-valued functions . . . . . . . . . . . . . . . 7.9 The filtration of an opeator function with respect to a contour . . 7.10 A general criterion for the existence of factorizations . . . . . . . . 7.11 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 240 240 242 245 251 259 267

Wiener-Hopf operators, Toeplitz operators and factorization 8.1 Holomorphic operator functions . . . . . . . . . . . . . . 8.2 Factorization of G ω (E)-valued functions . . . . . . . . . 8.3 The space L2 (Γ, H) . . . . . . . . . . . . . . . . . . . . . 8.4 Operator functions with values acting in a Hilbert space 8.5 Functions close to the unit operator or with positive real 8.6 Block T¨ oplitz operators . . . . . . . . . . . . . . . . . . 8.7 The Fourier transform of L1 (R, E) . . . . . . . . . . . . 8.8 The Fourier isometry U of L2 R, H . . . . . . . . . . . 8.9 The isometry V from L2 (T, H) onto L2 (R, H) . . . . . . 8.10 The algebra of operator functions L(H) ⊕ L1 R, L(H) 8.11 Factorization with respect to the realline . . . . . . . . 8.12 Wiener-Hopf integral operators in L2 [0, ∞[, H . . . . . 8.13 An example . . . . . . . . . . . . . . . . . . . . . . . . . 8.14 Comments . . . . . . . . . . . . . . . . . . . . . . . . . .

269 269 273 276 287 291 297 305 313 317 320 324 325 338 340

. . . . . . . . . . . . part . . . . . . . . . . . . . . . . . . . . . . . . . . .

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219 220 222 230

x 9

Contents Multiplicative cocycles with restrictions (F-cocycles) 9.1 F-cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The main results on cocycles with restrictions. Formulation reduction to OD,Z,m . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Cartan lemma with restrictions . . . . . . . . . . . . . . 9.4 Splitting over simply connected open sets after shrinking . . . 9.5 Runge approximation on simply connected open sets . . . . . 9.6 Splitting over simply connected open sets without shrinking . 9.7 Runge approximation. The general case . . . . . . . . . . . . 9.8 The Oka-Grauert principle . . . . . . . . . . . . . . . . . . . . 9.9 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . and . . . . . . . . . . . . . . . . . . . . . . . .

10 Generalized interpolation problems 10.1 Weierstrass theorems . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Right- and two-sided Weierstrass theorems . . . . . . . . . . . . 10.3 Weierstrass theorems for G ∞ (E)- and G ω (E)-valued functions . 10.4 Holomorphic G ∞ (E)-valued functions with given principal parts the inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . of . . . .

11 Holomorphic equivalence, linearization and diagonalization 11.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Linearization by extension and equivalence . . . . . . . . . . . . 11.3 Local equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 A theorem on local and global equivalence . . . . . . . . . . . . . 11.5 The finite dimensional case . . . . . . . . . . . . . . . . . . . . . 11.6 Local and global equivalence for finite meromorphic Fredholm functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Global diagonalization of finite meromorphic Fredholm functions 11.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

343 343 346 347 352 354 358 362 365 367 369 369 371 374 377 378 379 379 380 386 392 394

. 399 . 406 . 411

Bibliography

413

Index

419

Preface This is a book on holomorphic operator functions of a single variable and their applications, which is focussed on the relations between local and global theories. It is based on methods and technics of Complex analysis of scalar and matrix functions of several variables. The applications concern: interpolation, holomorphic families of subspaces and frames, spectral theory of polynomials with operator coefficients, holomorphic equivalence and diagonalization, and Plemelj-Muschelishvili factorization. The book also contains a theory of Wiener-Hopf integral equations with operator-valued kernels and a theory of infinite T¨ oplitz matrices with operator entries. We started to work on these topics long ago when one of us was a Ph.D. student of the other in Kishinev (now Cisinau) University. Then our main interests were in problems of factorization of operator-valued functions and singular integral operators. Working in this area, we realized from the beginning that different methods and tools from Complex analysis of several variables and their modifications are very useful in obtaining results on factorization for matrix and operator functions. We have in mind different methods and results concerning connections between local and global properties of holomorphic functions. The first period was very fruitful and during it we obtained the basic results presented in this book. Then World Politics started to interfere in our joint work in the new area. For a long time the authors became separated. One emigrated to Israel, the other was a citizen of East Germany, and the authorities of the second country prevented further meetings and communications of the authors. During that time one of us became more and more involved in Complex analysis of several variables and finally started to work mainly in this area of mathematics. Our initial aims were for a while frozen. Later the political situation in the world changed and after the reunification of Germany the authors with pleasure continued the old projects. During the time when our projects were frozen, the scientific situation changed considerably. There appeared in the literature new methods, results and applications. In order to cover the old and new material entirely in a modern form and terminology we decided to write this book. As always happens in such cases, during the writing new problems and gaps appear, and the material requires inclusion of additional material with new chapters containing new approaches, new results and plenty of unification and polishing. This work was done by the authors.

xii

Preface

We hope the book will be of interest to a number of large groups of experts in pure and applied mathematics as well as for electrical engineers and physicists. During the work on the book we obtained support of different kinds for our joint activities from the Tel-Aviv University and its School of Mathematical Sciences, the Family of Nathan and Lilly Silver Foundation, the Humboldt Foundation, the Deutsche Forschungsgemeinschaft and the Humboldt University in Berlin and its Institute of Mathematics. We would like to express our sincere gratitude to all these institutions for support and understanding. We would also like to thank the Faculty of Mathematics and Computer Sciences of the Kishinev University and the Institute of Mathematics and Computer Center of the Academy of Sciences of Moldova, where the work on this book was started.

Berlin, Tel-Aviv, November 2008

The authors

Introduction The book. This book contains a theory and applications of operator-valued holomorphic functions of a single variable. (By operators we always mean bounded linear operators between complex Banach spaces.) The applications concern some important problems on factorization, interpolation, diagonalization and others. The book also contains a theory of Wiener-Hopf integral equations with operatorvalued kernels and a theory of infinite T¨ oplitz matrices with operator entries. Our main attention is focussed on the connection between local and global properties of holomorphic operator functions. For this aim, methods from Complex analysis of several variables are used. The exposition of the material appears in style and terms of the latter field. Multiplicative cocycles. Grauert’s theory. The theory of multiplicative cocycles plays a central role in this book. It is a special case of the very deep and powerful theory of cocycles (fiber bundles) on Stein manifolds (any domain in C is a Stein manifold), which was developed in the 1950s by H. Grauert for cocycles with values in a (finite dimensional) complex Lie group. This theory then was generalized into different interesting directions. In 1968, L. Bungart obtained it for cocycles with values in a Banach Lie group, for example, the group of invertible operators in a Banach space. One of the main statements of Grauert’s theory is a principle which is now called the Oka-Grauert principle. Non-rigorously, this principle can be stated as follows: If a holomorphic problem on a Stein manifold has no topological obstructions, then it has a holomorphic solution. This important principle was first discovered in 1939 by K. Oka in the case of scalar functions. For domains in the complex plane C, Grauert’s theory is much easier but still not simple. It is even not simple for the case of cocycles with values in the group of invertible complex n × n-matrices when no topological obstructions appear. For operators in infinite dimensional Banach spaces, we meet essential difficulties, which are due to the fact that the group of invertible operators in a Banach space need not be connected. This becomes a topological obstruction if the domain in C is not simply connected. So, for operator functions, the Oka-Grauert principle is meaningful also for domains in C.

xiv

Introduction

For the problem of Runge approximation, the Oka-Grauert principle claims the following: Runge approximation of a holomorphic invertible operator function by holomorphic invertible functions is possible if this is possible by continuous invertible functions. From this it follows that such a Runge approximation always holds when the domain is simply connected or the group of invertible operators is connected. The latter is the case for the group of invertible operators in a Hilbert space, and in particular, for the group of invertible complex n × n-matrices. For simply connected domains, the proof of the Runge approximation theorem for invertible operator functions is not difficult and can be obtained without the theory of cocycles. We show this at the end of Chapter 2. For general domains however, this proof is much more difficult (even in the case of matrix-valued functions) and will be given only in Chapter 5 in the framework of the theory of multiplicative cocycles. A special type of multiplicative cocycles is given by two open sets D1 and D2 in C and an invertible holomorphic operator function on D1 ∩ D2 . For this type, the following is proved: 0.0.1 Theorem. Let E be a Banach space, let GL(E) be the group of invertible operators in E, let D1 , D2 ⊆ C be two open sets, and let A : D1 ∩ D2 → GL(E) be holomorphic. Assume that at least one of the following two conditions is satisfied: (i) The union D1 ∪ D2 is simply connected. (ii) All values of A belong to the same connected component of GL(E). Then there exist holomorphic operator functions Aj : Dj → GL(E), j = 1, 2, such that on D1 ∩ D2 . (0.0.1) A = A1 A−1 2 If both topological conditions (i) and (ii) in Theorem 0.0.1 are violated, then the assertion of Theorem 0.0.1 is not true. A simple counterexample will be given in Section 5.6.2 for the case when D1 ∪ D2 is an annulus. The following operator version of the Weierstrass product theorem (on the existence of holomorphic functions with given zeros) is a straightforward consequence of Theorem 0.0.1. 0.0.2 Theorem. Let E be a Banach space, let GL(E) be the group of invertible operators in E, and let GLI (E) be the connected component in GL(E) which contains the unit operator I. Let D ⊆ C be an open set and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, a neighborhood Uw ⊆ D of w with Uw ∩ Z = {w} and a holomorphic operator function Aw : Uw \ {w} → GL(E) are given. Further assume that at least one of the following two conditions is fulfilled: (i) The set D is simply connected. (ii) The values of each Aw , w ∈ Z, belong to GLI (E).

Introduction

xv

Then there exist a holomorphic operator function B : D\Z → GL(E) and a family of holomorphic operator functions Hw : Uw → GL(E) such that Hw Aw = B

on Uw \ {w}, ,

w∈Z.

The classical Weierstrass product theorem we get for E = C and Hw (z) = (z − w)κw , κw ∈ N∗ . There are also a “right-sided” and a “two-sided” version of Theorem 0.0.2. Contents. The book consists of an introduction and eleven chapters. Let us now describe in more detail the content of each chapter separately. The first chapter contains the generalization to functions with values in Banach spaces of the traditional material from Complex analysis of one variable which is usually contained in the beginning of a basic course. Chapter 2 starts with Pompeiju’s integral formula for solutions of the inhomogeneous Cauchy-Riemann equation, the Runge approximation theorem, the Mittag-Leffler theorem, and the Weierstrass product theorem. Then, in Sections 2.6 and 2.7, we present the (less well known) “Anschmiegungsatz” of Mittag-Leffler and a strengthening of the Weierstrass product theorem. In the case of the Weierstrass product theorem and its generalization, in this chapter, we still restrict ourselves to scalar functions. It is one of the main goals of this book, to generalize these results to the case of operator functions, using Grauert’s theory of cocycles. Chapter 3 is dedicated to the splitting problem with respect to a contour for functions with values in a Banach space, as well as to the factorization problem for scalar functions with respect to a contour. In Chapter 4 we generalize to finite meromorphic Fredholm operator functions the classical Rouché theorem from Complex analysis and the Smith factorization form. The proof is based on the local Smith form. Chapter 5 is entirely dedicated to the theory of multiplicative cocycles, which were discussed in large before. Chapter 6 contains a theory of families of subspaces of a Banach space E. First we introduce a complete metric on the set G(E) of closed subspaces of E, the so-called gap metric. A continuous family of subspaces of E then will be defined as a continuous function with values in G(E), and a holomorphic family of subspaces of E will be defined as a continuous family of subspaces which is locally the image of a holomorphic operator function. Vector functions with values in such a family are called sections of the family. Note that we do not require that the members of a holomorphic family be complemented in the ambient space. It may even happen they are not pairwise isomorphic. An example is given in Section 6.5. First we prove the following results: any additive cocycle of holomorphic sections in a holomorphic family of subspaces splits; for any holomorphic operator function A whose image is a holomorphic family of subspaces, and any holomorphic section f of this family, there exists a global holomorphic vector function u that solves the equation Au = f ; for any holomorphic family of subspaces there exists a global holomorphic operator function with this family as image. Proving this,

xvi

Introduction

the main difficulty is the solution of certain local problems (in this generality, published for the first time in this book). In terms of Complex analysis of several variables, the solution of these local problems means that any holomorphic family of subspaces is a so-called Banach coherent sheaf (a generalization of the notion of coherent sheaves). After solving this we proceed by standard methods that are well-known in Complex analysis of several variables. Then we consider holomorphic families of subspaces, which we call injective and which have the additional property that, locally, the family can be represented as the image of a holomorphic operator function with zero kernel. We study the problem of a corresponding global representation. Here we need the theory of multiplicative cocycles from Chapter 5. It turns out that this is not always possible, but we have again an Oka-Grauert principle. Then we study holomorphic families of complemented subspaces (which are injective), where we can prove more precise results than for arbitrary injective families. Again there is an Oka-Grauert principle. At the end we consider the special case of families of subspaces which are finite dimensional or of finite codimension. Here there are no topological restrictions. Chapters 7 and 8 are dedicated to factorization of operator functions with respect to a contour and the connection with Wiener-Hopf and T¨ oplitz operators. This type of factorization was in fact considered for the first time in the pioneering works of Plemelj and of Muschelishvili. Because of that we call it PlemeljMuschelishvili factorization. We start with the local principle, which quickly follows from the theory of multiplicative cocycles and which allows us to prove theorems on factorization for different classes of operator functions. The local principle reduces the problem to functions which are already holomorphic in a neighborhood of the contour. For further applications we need a generalization of the theory of multiplicative cocycles. This is the topic of Chapter 9, where we introduce cocycles with restrictions. Let us offer an example (which is basic for all cocycles with restrictions). Suppose that in Theorem 0.0.1 an additional set Z ⊆ D1 ∪ D2 , discrete and closed in D, and positive integers mw , w ∈ Z, are given. Assume that the function A − I has a zero of order mw at each w ∈ D1 ∩ D2 ∩ Z. Then the theory of cocycles with restrictions gives the additional information that the functions A1 and A2 in Theorem 0.0.1 can be chosen so that, for all w ∈ Dj ∩ Z, j = 1, 2, the function Aj − I has a zero of order mw at w. In Chapter 10, by means of the theory of cocycles with restrictions, we essentially improve the Weierstrass product Theorem 0.0.2: The functions Hw in this theorem now can be chosen so that, additionally, for each w ∈ Z, the function Hw − I has a zero of an arbitrarily given order mw at w. This has different consequences that are discussed in this short chapter. Chapter 11 is dedicated to holomorphic equivalence and its applications to linearization and diagonalization. Let E be a Banach space, let L(E) be the space of bounded linear operators in E, let GL(E) be the group of invertible operators from L(E), let D ⊆ C be an open set, and let Z be a discrete and closed subset

Introduction

xvii

of D. Then two holomorphic operator functions A, B : D \ Z → L(E) are called (globally) holomorphically equivalent over D if there exist holomorphic operator functions S, T : D → GL(E) such that A = SBT on D. In the first section, results are presented that explain the importance of the notion of holomorphic equivalence in spectral theory of linear operators and holomorphic operator functions. It contains the following two results: 1) For each relatively compact open subset Ω of D, each holomorphic operator function A : D → L(E), after an appropriate extension, becomes holomorphically equivalent to a function of the form zI − T , z ∈ Ω, where T is a constant operator and I is the identical operator (Theorem 11.2.1). 2) Two operators T, S ∈ L(E) with the spectra σ(A) and σ(B) are similar if and only if some extensions of the functions zI − T and zI − S are holomorphically equivalent over some neighborhood of σ(A) ∪ σ(B) (Corollary 11.2.3). The remainder of this section is devoted to the relation between global and local holomorphic equivalence where two holomorphic operator functions are called locally holomorphically equivalent if, for each point, they are holomorphically equivalent over some neighborhood of this point. We prove that two meromorphic operator functions with meromorphic inverse are locally holomorphically equivalent if and only if they are globally holomorphically equivalent (Theorem 11.4.2), and we prove that any finite meromorphic Fredholm operator function is globally holomorphically equivalent to a diagonal function (Theorem 11.7.6). The local fact behind this is the Smith representation of matrices of germs of scalar holomorphic functions. Acknowledgement. In the beginning of the 1970s, on an invitation of one of us, M.A. Shubin visited Kishinev and gave two talks about applications of Grauert’s theory and the theory of coherent analytic sheaves to different results for linear operators. One of the talks was on the local principle for Plemelj-Muschelishvili factorization of matrix functions and the second was about the analysis of holomorphic families of subspaces. These talks had on us an important influence. Very soon after this we came up with a series of papers on operator-valued cocycles in the case of one variable with new direct proofs and also with new results and applications to operator functions. At the end this development led to this book. It is our pleasure to thank M.A. Shubin providing us with the initial input.

Notation Here we give a list of standard symbols and some remarks concerning the terminology used in this book without further explanation: – C is the complex plane, R is the real axis, C∗ := C \ {0}, R∗ := R \ {0}. – N is the set of natural numbers (including 0), N∗ := N \ {0}. – Z is the set of entire numbers. – Banach spaces and Banach algebras are always complex. – If E, F are Banach spaces, then we denote by L(E, F ) the Banach space of bounded linear operators operators from E to F , endowed with the operator norm. We set L(E) = L(E, E), and we denote by GL(E) the group of all invertible operators from L(E). By a projector in E we always mean an operator P ∈ L(E) with P 2 = P . – By an operator we always mean a bounded linear operator between two Banach spaces. – Let E, F be Banach spaces, and let A ∈ L(E, F ). Then we denote by Im A the image, and by Ker A the kernel of A. The operator A is called injective, if Ker A = {0}, and it is called surjective if Im A = F . – The unit operator of a Banach space E will be denoted by I or IE . – For n ∈ N∗ we denote by L(n, C) the algebra of complex n × n matrices, and by GL(n, C) we denote the group of invertible elements of L(n, C). – By a neighborhood we always mean an open neighborhood, if not explicitly stated to be anything else. – If U is a set in a topological space X, then U always denotes the topological closure of U in X (and not the complement).

xx

Notation – By C 0 -functions or functions of class C 0 we mean continuous functions. If Γ is a subset of C and M is a subset of a Banach space, then we denote by C M (Γ) or by (C 0 )M (Γ) the set of all continuous functions f : Γ → M . – If U ⊆ C is an open set, U = ∅, and E is a Banach space, then a function f : U → E is called C k or of class C k on U , k ∈ N∗ ∪ {∞}, if it is k times continuously differentiable with respect to the canonical real coordinates of C. – If U ⊆ C is an open set, U = ∅, and M is a subset of a Banach space E, then we denote by (C k )M (U ) the set of all Ck -functions f : U → E such that f (z) ∈ M for all z ∈ U , and by OM (U ) we denote the set of all holomorphic (Def. 1.1.1) functions f : U → E such that f (z) ∈ M for all z ∈ U . ∗

– We set O(U ) = OC (U ), C k (U ) = (C k )C (U ) and O∗ (U ) = OC (U ) for each open U ⊆ C and k ∈ N. – If K ⊆ C is a (not necessarily open) set of uniqueness for holomorphic functions (for example, the closure of an open set, or an interval) and E is a Banach space, then we speak also about a holomorphic function f : K → E to say that f is the restriction of an E-valued holomorphic function defined in a neighborhood of K. – If D ⊆ C is an open set with piecewise C 1 -boundary (Def. 1.4.1), then we denote by ∂D the boundary of D endowed with the orientation defined by D (Sect. 1.4.1), i.e., D lies on the left side of ∂D.

Chapter 1

Elementary properties of holomorphic functions This chapter is devoted to the basic facts usually contained in a basic course on Complex analysis of one variable. The difference is that we do this for functions with values in a Banach space. Many (not all) of these results will be deduced by the Hahn-Banach theorem from the corresponding scalar fact. Some care is necessary with respect to the maximum principle. The strong version, that the norm of a non-constant holomorphic function does not admit local maxima, is not true in general. For example, it fails for l∞ and it is true for Hilbert spaces.

1.1

Definition and first properties

The notion of a holomorphic function with values in a Banach space can be defined as in the scalar case by complex differentiability: 1.1.1 Definition. Let E be Banach space, and let U ⊆ C be an open set. A function f : U → E is called complexly differentiable or holomorphic if, for each w ∈ U , f (w) := lim

z→w

f (z) − f (w) z−w

exists. Clearly, then also the partial derivatives of f with respect to the canonical real coordinates x, y exist, and the Cauchy-Riemann equation holds: f (w) =

∂f ∂f (w) = i (w) , ∂x ∂y

w ∈ D.

(1.1.1)

The function f : U → E, which is then defined, will be called the complex derivative or simply the derivative of f .

2

Chapter 1. Elementary properties of holomorphic functions

The space of all holomorphic functions from U to E will be denoted by OE (U ). 1.1.2. From this definition the following facts follow immediately: – Each holomorphic function with values in a Banach space is continuous.1 – If U ⊆ C is open, E is a Banach space, f, g ∈ OE (U ) and α, β ∈ OC (U ), then αf + βg ∈ OE (U ), and (αf + βg) = α f + αf + β g + βg on U . – If U ⊆ C is open, A is a Banach algebra, and f, g ∈ OA (U ), then f g ∈ OA (U ) and (f g) = f g + f g on U . – If U ⊆ C is open, E, F are Banach spaces, f ∈ OE (U ), and A ∈ OL(E,F ) (U ), then Af ∈ OF (U ) and (Ag) = A g + Ag on U . – If U ⊆ C is open, A is a Banach algebra with unit, GA is the group of invertible elements of A, and f : U → GA is holomorphic, then f −1 is holomorphic and (f −1 ) = −f −1 f f −1 on U . – If U, V ⊆ C are open, E is a Banach space, and α : U → V , f : V → E are holomorphic, then the composition f ◦ α is holomorphic and (f ◦ α) = α (f ◦ α)

on U.

(1.1.2)

– If I ⊆ R is an interval, U ⊆ C is open, E is a Banach space, α : I → U is differentiable, and f : U → E is holomorphic, then the composition f ◦ α is differentiable on I and (f ◦ α) = α (f ◦ α)

on I.

(1.1.3)

The theorem on uniqueness of holomorphic functions is deeper, but by means of the Hahn-Banach theorem it can be quickly obtained from the scalar fact: 1.1.3 Theorem. Let D ⊆ C be a connected open set, and let zn ∈ D, n ∈ N∗ , be a sequence which converges to a point z0 ∈ D such that zn = z0 for all n ∈ N∗ . Further let E be a Banach space, and let f, g : D → E be two holomorphic functions such that f (zn ) = g(zn ) for all n ∈ N∗ . Then f ≡ g on D. Proof. Let E be the dual of E. Then it follows from the theorem on uniqueness of scalar holomorphic functions that, for all Φ ∈ E , Φ ◦ f ≡ Φ ◦ g. By the HahnBanach theorem this implies that f ≡ g. The same is true for Liouville’s theorem: 1.1.4 Theorem. Let E be a Banach space, let E be the dual of E, and let f : C → E be a holomorphic function. Suppose, for each Φ ∈ E , the function Φ ◦ f is bounded on C (which is the case, for example, if the function f is bounded). Then f is constant. 1 They

are even of class C ∞ , but, as in the scalar case, this can be proved only after the Cauchy formula is obtained.

1.2. The maximum principle

3

Proof. It follows from Liouville’s theorem for scalar holomorphic functions that Φ ◦ f is constant for all Φ ∈ E . By the Hahn-Banach theorem this implies that f is constant.

1.2

The maximum principle

The following (weak) version of the maximum principle again can be obtained by means of the Hahn-Banach theorem immediately from the maximum principle for scalar holomorphic functions: 1.2.1 Theorem. Let D ⊆ C be a bounded open set, let E be a Banach space, and let f : D → E be a continuous function which is holomorphic in D. Denote by ∂D the boundary of D. Then max f (z) = max f (z) . z∈D

z∈∂D

(1.2.1)

Proof. Let z0 be an arbitrary point in D, and let E be the dual of E. Then, for each Φ ∈ E , the function Φ ◦ f is holomorphic and hence, by the maximum principle for scalar holomorphic functions, (Φ f (z0 ) ≤ max (Φ ◦ f )(z) ≤ Φ max f (z) . z∈∂D

z∈∂D

By the Hahn-Banach theorem, this implies |f (z0 )| ≤ max f (z) . z∈∂D

As z0 was chosen arbitrarily in D, this implies (1.2.1).

The strong maximum principle “If a holomorphic function, defined on a connected open set, admits a local maximum, then it is constant” is not true for functions with values in an arbitrary Banach space. Indeed, take the space C2 with the norm (ξ1 , ξ2 ) max = max{|ξ1 |, |ξ2 |} and consider the holomorphic function f (z) = (z, 1) defined for |z| < 1. Clearly f is not constant but

f max ≡ 1. For functions with values in a Hilbert space we have the strong maximum principle: 1.2.2 Theorem. Let D ⊆ C be a connected open set, let H be a Hilbert space, and let f : D → H be a holomorphic functions such that, for some z0 ∈ D and ε > 0,

f (z0 ) ≥ f (z) Then f is constant.

for all |z − z0 | ≤ ε.

(1.2.2)

4


To prove this we need some facts on scalar holomorphic functions, which are not necessarily contained in a standard course on Complex analysis. We therefore first present these facts with proofs. 1.2.3 Lemma. Let D ⊆ C be an open set, and let ϕ : D → C be a scalar holomorphic function. Let x, y be the canonical real coordinates on C, and let Δ :=

∂2 ∂2 + 2 2 ∂x ∂y

be the Laplace operator. Then Δ|ϕ|2 = 4|ϕ |2 .

(1.2.3)

Proof. We have ∂2 ∂2 ∂ ∂ϕ ∂ 2 ϕ ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∂2ϕ 2 |ϕ| = (ϕϕ) = ϕ + ϕ = +ϕ 2 + +ϕ 2 2 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x and, in the same way, ∂2 ∂ϕ ∂ϕ ∂ 2 ϕ ∂ϕ ∂ϕ ∂2ϕ 2 |ϕ| = ϕ + . + + ϕ ∂y 2 ∂y ∂y ∂y 2 ∂y ∂y ∂y 2 Since ϕ is holomorphic and therefore, by the Cauchy-Riemann equation, ϕ =

∂ϕ ∂ϕ =i ∂x ∂y

and ϕ =

∂ϕ ∂ϕ = −i , ∂x ∂y

this implies ∂2ϕ ∂2ϕ ∂2ϕ ∂2ϕ ∂2 2 2 |ϕ| = ϕ ϕ + ϕ + ϕ ϕ + ϕ = 2|ϕ | + ϕ + ϕ ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 and, in the same way, ∂2 ∂2ϕ ∂2ϕ 2 2 |ϕ| = 2|ϕ | + ϕ + ϕ . ∂y 2 ∂y 2 ∂y 2 Hence

∂2ϕ ∂2ϕ + Δ|ϕ| = 4|ϕ | + ϕ ∂x2 ∂y 2 2

2

∂2ϕ ∂2ϕ +ϕ + ∂x2 ∂y 2

= 4|ϕ |2 + ϕΔϕ + ϕΔϕ.

Since Δϕ = Δϕ = 0 (real and imaginary part of ϕ are harmonic), this implies (1.2.3).

1.2. The maximum principle

5

1.2.4 Lemma. Let r0 > 0, and let ϕ be a scalar holomorphic function defined on the disc |z| < r0 . Set 1 M (r) = 2π

2π it 2 ϕ re dt

for 0 ≤ t < r0 .

(1.2.4)

0

Then M is of class C

∞

on [0, r0 [, and 2 |ϕ |2 dλ M (r) = πr

for 0 < r < r0 .

(1.2.5)

|z| 0 such that the closed disc |z − z0 | ≤ r is contained in D, ρ(z0 ) ≤

1 2π

2π ρ z0 + reit dt. 0

8


Therefore the maximum principle stated in Theorem 1.2.1 can be viewed also as a consequence of the maximum principle for subharmonic functions.

1.3

Contour integrals

Here we collect a number of definitions for later reference. 1.3.1 Definition (C 1 -contours). A set Γ ⊆ C is called a connected C 1 -contour if there exist real numbers a < b and a C 1 -function γ : [a, b] → C with Γ = γ [a, b] such that: (i) γ (t) = 0 for all a ≤ t ≤ b; (ii) γ(t) = γ(s) for all a ≤ t, s < b with t = s; (iii) either γ(b) = γ(t) for all a ≤ t < b or γ(b) = γ(a) and γ (b) = γ (a). Then the function γ is called a C 1 -parametrization of Γ. If γ(b) = γ(a), then Γ is called closed. By a (not necessarily connected) C 1 -contour we mean the union of a finite number of pairwise disjoint connected C 1 -contours. 1.3.2 Definition (Piecewise C 1 -contours). A set Γ ⊆ C is called a connected piecewise C 1 -contour in each of the following three cases: (I) There exist real numbers a < b and a C 1 -function γ : [a, b] → C with Γ = γ [a, b] such that: (i) γ (t) = 0 for all a ≤ t ≤ b; (ii) γ(t) = γ(s) for all a ≤ t, s < b with t = s; (iii) γ(b) = γ(a) and

γ (a) γ (b)

∈ C\] − ∞, 0].3

(II) There exist finitely many real numbers a = t1 < . . . < tm = b and a continuous function γ : [a, b] → C with Γ = γ([a, b]) such that: (i) For each 1 ≤ j ≤ m − 1, the function γj := γ

[tj ,tj+1 ]

is of class C 1 on [tj , tj+1 ] and γj (t) = 0 for all tj ≤ t ≤ tj+1 . (ii)

γj (tj+1 ) γj+1 (tj+1 )

∈ C\] − ∞, 0] for 1 ≤ j ≤ m − 2.

(iii) γ(t) = γ(s) for all a ≤ t, s ≤ b with t = s. 3 i.e.,

either Γ is smooth at γ(b) = γ(a) or Γ forms a non-zero angle at γ(b) = γ(a).

1.3. Contour integrals

9

(III) If in case (II) condition (iii) is replaced by (iii ) γ(t) = γ(s) for all a ≤ t, s < b with t = s, γ(b) = γ(a) and C\] − ∞, 0].

γ (a) γ (b)

∈

The contour Γ is called closed, if and only if, γ(b) = γ(a). The function γ then is called a piecewise C 1 -parametrization of Γ. By a (not necessarily connected) piecewise C 1 -contour we mean the union of a finite number of pairwise disjoint connected piecewise C 1 -contours. Such a contour is called closed if each connected component is closed. 1.3.3 Definition (Orientation of a contour). First let Γ ⊆ C be a connected piecewise C 1 -contour. If γ : [a, b] → C and γ ∗ : [a∗ , b∗ ] → C are two piecewise C 1 -parametrizations of Γ, then by definition of a piecewise C 1 -parametrization, on [a∗ , b∗ [, the function γ −1 ◦ γ ∗ is well defined, and, as it is continuous, it is either strictly monotonicly increasing or strictly monotonicly decreasing. In the first case we call γ and γ∗ equivalent. In this way the set of all piecewise C 1 -parametrizations of Γ is divided into two equivalence classes. We say that Γ is oriented if one of these two equivalence classes is chosen. If this is done, then a piecewise C 1 -parametrization of Γ is called positively oriented if it belongs to the chosen class. A (not necessarily connected) piecewise C 1 -contour Γ is called oriented if on each connected component of Γ an orientation is chosen. 1.3.4 Definition. Let Γ be an oriented piecewise C 1 -contour, let E be a Banach space, and let f : Γ → E be a continuous function. If Γ is connected and γ : [a, b] → C is a positively oriented, piecewise C 1 parametrization of Γ, then it follows from the substitution rule (which follows as in the scalar case from the chain rule) that

b f (z)dz :=

f γ(z) γ (t)dt

(1.3.1)

a

Γ

is independent of the choice of γ. Therefore, by (1.3.1) an integral Γ f (z)dz is well defined. If Γ is not connected and Γ1 , . . . , Γn are the connected components of Γ, then we define n f (z)dz = f (z)dz. (1.3.2) Γ

j=1 Γ

j

10


If Γ is a circle of radius r centered at z0 ∈ C and if Γ is oriented by the parametrization γ(t) := z0 + reit , 0 ≤ t ≤ 2π, then we define

2π

f (z)dz :=

|z−z0 |=r

f (z)dz = ir

f z0 + reit eit dt.

(1.3.3)

0

Γ

1.3.5 Definition. Let Γ be a piecewise C 1 -contour. If Γ is connected and γ : [a, b] → C is a piecewise C 1 -parametrization of Γ, then it follows from the substitution rule (which follows as in the scalar case from the chain rule) that b |Γ| := |γ (t)|dt (1.3.4) a

is independent of the choice of γ. Therefore, by (1.3.4) a number |Γ| is well defined. If Γ is not connected and Γ1 , . . . , Γn are the connected components of Γ, then we define n Γj . |Γ| = (1.3.5) j=1

The number |Γ| is called the length of Γ. 1.3.6 Proposition. Let Γ be a piecewise C 1 -contour, let E be a Banach space, and let f : Γ → E be a continuous function. Then f (z)dz ≤ |Γ| max f (z) . z∈Γ

(1.3.6)

Γ

Proof. We may assume that Γ is connected. Let γ : [a, b] → C be an oriented, piecewise C 1 -parametrization of Γ, and let E be the dual of E. Then, for each Φ ∈ E with Φ = 1, b

b f (z)dz = Φ f γ(t) γ (t)dt = Φ f γ(t) γ (t)dt Φ a

Γ

a

b b ≤ Φ f γ(t) γ (t)dt ≤ max f γ(t) |γ (t)|dt = |Γ| max f (z) . z∈Γ a≤t≤b a

a

By the Hahn-Banach theorem, this implies (1.3.6).

1.4. The Cauchy integral theorem

1.4

11

The Cauchy integral theorem

1.4.1. Let D ⊆ C be an open set. We shall say that D has a piecewise C 1 -boundary if the boundary of D (in C) is a closed piecewise C 1 -contour Γ (Def. 1.3.2) such that each point of Γ is also a boundary point of C \ D. Let D ⊆ C be an open set with piecewise C 1 -boundary. Then different orientations of Γ are possible (more than two if Γ is not connected). One of these orientations is of particular interest: The orientation defined by D: This is the orientation of Γ such that D is “on the left” of Γ with respect to this orientation.4 In this case, we also say that Γ is oriented by D or oriented as the boundary of D. By ∂D we denote the boundary of D if it is oriented by D. 1.4.2 Theorem (Cauchy integral theorem). Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, let E be a Banach space, and let f : D → E be a continuous function which is holomorphic in D. Then f (z)dz = 0. (1.4.1) Γ

Proof. Let E be the dual of E. Then, for each Φ ∈ E , Φ ◦ f is a scalar function which is continuous on D and holomorphic in D. Hence, by the Cauchy integral theorem for scalar functions, f (z)dz = Φ f (z) dz = 0 for all Φ ∈ E . (1.4.2) Φ ∂D

∂D

By the Hahn-Banach theorem this implies (1.4.1).

1.4.3. Recall that an open set D ⊆ C is called simply connected if it is connected and, for any continuous function γ : [0, 1] → D with γ(0) = γ(1), there exists a continuous function H : [0, 1] × [0, 1] −→ D such that H(0, t) = H(1, t) for all 0 ≤ t ≤ 1, H(·, 0) = γ for all 0 ≤ s ≤ 1 and H(·, 1) is a constant. We need also the following homotopy version of the Cauchy integral theorem for functions with values in a Banach space. 4 A possible formal definition: We say that D is on the left of Γ if the following condition is satisfied: If γ : [a, b] → C is a positively oriented piecewise C 1 -parametrization of one of the connected components of Γ, then εiγ (t) ∈ D for each a ≤ t ≤ b such that γ(t) is a smooth point of Γ and any sufficiently small ε > 0.

12


1.4.4 Theorem. Let D ⊆ C be a simply connected open set, let E be a Banach space, let f : D → E be holomorphic, and let γ1 , γ2 : [0, 1] → D be two piecewise C 1 -functions with γ1 (0) = γ2 (0) and γ1 (1) = γ2 (1).5 Then 1

f γ1 (z) γ1 (z)dz =

0

1

f γ1 (z) γ2 (z)dz.

(1.4.3)

0

Proof. Let E be the dual of E. Then, by the homotopy version of the Cauchy integral theorem for scalar functions, for each Φ ∈ E , 1 Φ

f γ1 (z) γ1 (z)dz

0

1

= Φ f γ1 (z) γ1 (z)dz 0

1 1 = Φ f γ1 (z) γ2 (z) dz = Φ f γ2 (z)γ2 (z)dz . 0

0


1.5

The Cauchy formula

1.5.1 Theorem (Cauchy formula). Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, let E be a Banach space, and let f : D → E be a continuous function which is holomorphic in D. Then f (z) 1 f (w) = dz, w ∈ D. (1.5.1) 2πi z−w ∂D

Proof. Let w ∈ D be given. Let E be the dual of E. Then, for each Φ ∈ E , Φ ◦ f is a scalar function which is continuous on D and holomorphic in D. Hence, by the Cauchy formula for scalar functions, Φ f (z) f (z) 1 1 Φ f (w) = dz = Φ dz for all Φ ∈ E . 2πi z−w 2πi z−w ∂D

∂D


1.5.2 Lemma. Let Γ ⊆ C be an oriented, piecewise C 1 -contour (not necessarily closed), let E be a Banach space, and let f : Γ → E be continuous. Let n ∈ N∗ and set f (ζ) F (z) = dζ , z ∈ C \ Γ. (1.5.2) (ζ − z)n Γ

we do not assume that the images γ1 [0, 1] and γ2 [0, 1] are piecewise C 1 -contours in the sense of Definition 1.3.2. 5 Here

1.5. The Cauchy formula

13

Then F is holomorphic on C \ Γ and f (ζ) F (z) = n dζ , (ζ − z)n+1

z ∈ C \ Γ.

(1.5.3)

Γ

Moreover lim F (z) = 0.

|z|→∞

(1.5.4)

Proof. Using the estimate from Proposition 1.3.6, we get

F (z) ≤ |Γ| max ζ∈Γ

f (ζ) . |ζ − z|n

As Γ is compact and therefore lim max

|z|→∞ ζ∈Γ

1 = 0, |ζ − z|n

this implies (1.5.4). It remains to prove that F is holomorphic on C \ Γ. Let w ∈ C \ Γ be given. We must prove that 1 − 1 n (ζ−z)n (ζ−w)n lim f (ζ)dζ = 0. (1.5.5) − z→w z−w (ζ − w)n+1 Γ

We have 1 (ζ−z)n

= = = = =

−

1 (ζ−w)n

n − z−w (ζ − w)n+1 (ζ − w)n+1 − (ζ − z)n (ζ − w) − n(ζ − z)n (z − w) (z − w)(ζ − z)n (ζ − w)n+1 (ζ − z + z − w)n+1 − (ζ − z)n (ζ − z + z − w) − n(ζ − z)n (z − w) (z − w)(ζ − z)n (ζ − w)n+1 n+1 n+1 n+1−k (z − w)k − (ζ − z)n+1 − (n + 1)(ζ − z)n (z − w) k=0 k (ζ − z) (z − w)(ζ − z)n (ζ − w)n+1 n+1 n+1 n+1−k (z − w)k k=2 k (ζ − z) (z − w)(ζ − z)n (ζ − w)n+1 n+1 n+1 n+1−k (z − w)k−2 k=2 k (ζ − z) (z − w). (ζ − z)n (ζ − w)n+1

If ε > 0 is chosen so small that the disc |z − w| ≤ ε is contained in C \ Γ, this implies that, for some constant C < ∞, 1 1 (ζ−z)n − (ζ−w) n n ≤ C|z − w| if |z − w| < ε and ζ ∈ Γ. − n+1 z−w (ζ − w)

14


By the estimate from Proposition 1.3.6 this further implies 1 1 n (ζ−z)n − (ζ−w)n ≤ |Γ| max f (ζ) C|z − w| f (ζ)dζ − ζ∈Γ z−w (ζ − w)n+1 Γ

for |z − w| < ε, which proves (1.5.5).

In view of this lemma, the Cauchy formula immediately implies: 1.5.3 Corollary. Any holomorphic function with values in a Banach space is infinitely times complexly differentiable (Def. 1.1.1). In particular, it is of class C ∞ . Moreover, if D and f are as in Theorem 1.5.1 and if we denote by f (n) the n-th complex derivative of f in D, then f (z) n! (n) f (w) = dz, w ∈ D. (1.5.6) 2πi (z − w)n+1 ∂D

1.5.4 Theorem. Let D ⊆ C be an open set, and let M be a subset of D such that there exists a piecewise C 1 -contour in C with M ⊆ Γ. Further, let E be a Banach space, and let f : D → E be a continuous function which is holomorphic on D \M . Then f is holomorphic on D. Proof. Let w be an arbitrary point in M . Set Δε = z ∈ C |z − w| < ε

for ε > 0.

It is sufficient to prove that, for sufficiently small ε > 0, f is holomorphic on Δε . By hypothesis there exists a piecewise C 1 -contour Γ in C with M ⊆ Γ. Enlarging Γ if necessary, we may achieve that w is an “inner point” of Γ, i.e., that, for some ε0 > 0, the disc Δε0 is divided by Γ into two connected open sets Δ+ ε0 and Δ− . Moreover, we can choose 0 < ε < ε so small that the closed disc Δ is 0 ε ε0 contained in D, and the open sets + Δ+ ε := Δε ∩ Δε0

and

− Δ− ε := Δε ∩ Δε0

− have piecewise C 1 -boundaries ∂Δ+ ε and ∂Δε . Then, by Cauchy’s formula and by Cauchy’s integral theorem, f (z) if z ∈ Δ+ 1 f (ζ) ε , dζ = 2πi ζ −z 0 if z ∈ Δ− ε ∂Δ+ ε

and 1 2πi

∂Δ− ε

f (ζ) dζ = ζ −z

f (z) 0

if z ∈ Δ− ε , if z ∈ Δ+ ε .

1.6. The Hahn-Banach criterion

15

Since the orientation of Γ ∩ Δε as a part of ∂Δ+ ε is different from its orientation as a part of ∂Δ− , this implies that ε f (ζ) 1 dζ = f (z) if z ∈ Δε \ Γ. 2πi ζ −z |ζ−w|=ε

Since f is continuous on Δε , this further implies that f (ζ) 1 dζ = f (z) for all z ∈ Δε . 2πi ζ −z |ζ−w|=ε

In view of Lemma 1.5.2 this proves that f is holomorphic on Δε .

1.6

The Hahn-Banach criterion

1.6.1 Theorem (Hahn-Banach criterion). Let D ⊆ C be an open set, let E be a Banach space, let E be the dual of E, and let f : D → E be a function. Then the following two conditions are equivalent: (i) The function f is holomorphic on D. (ii) For each Φ ∈ E , the scalar function Φ ◦ f is holomorphic on D. Proof. (i) ⇒ (ii) is obvious. Suppose (ii) is satisfied. We first prove that then f is continuous. Consider an arbitrary point z0 ∈ D and choose r > 0 so small that the disc |z − z0 | ≤ r is contained in D. Since, for each Φ ∈ E , Φ◦f is holomorphic on D and, hence, Φ◦f is bounded on |z −z0 | ≤ r, the set Φ f (z) |z − z0 | ≤ r is bounded for each Φ ∈ E . Since weakly bounded sets are strongly bounded, it follows that the set {f (z) | |z − z0 | ≤ r} is bounded in E, i.e., we have a constant C < ∞ such that

f (z) ≤ C for all |z − z0 | ≤ r. (1.6.1) Now let zn , n ∈ N∗ , be a sequence which converges to z0 such that |zn − z0 | < r for all n ∈ N∗ . From the Cauchy formula, the estimate from Proposition 1.3.6 and estimate (1.6.1) then it follows that, for all Φ ∈ E ,

Φ f (z0 ) − f (zn ) = 1 2π ≤ r max Φ f (z) |z−z0 |=r

|z−z0 |=r

Φ f (z)

1 1 − z − z0 z − zn

dz

1 1 1 1 ≤ r Φ C max . − − z − z0 z − zn z − zn |z−z0 |=r z − z0

16


By the Hahn-Banach theorem this implies that f (z0 ) − f (zn ) ≤ rC max |z−z0 |=r

1 1 − . z − z0 z − zn

Hence limn→∞ f (zn ) = f (z0 ). To prove that f is holomorphic, we again consider an arbitrary point z0 ∈ D and choose r > 0 so small that the disc |z − z0 | ≤ r is contained in D. As f is continuous, by Lemma 1.5.2 the function 1 f (ζ) F (z) := dζ , |z − z0 | < r, 2πi ζ −z |ζ−z0 |=r

is holomorphic. Therefore it remains to prove that f (z) = F (z) for |z − z0 | < r. Let such z be given. Since, for each Φ ∈ E , the function Φ ◦ f is holomorphic, it follows from the Cauchy formula and then from the definition of F that

Φ f (ζ) 1 f (ζ) 1 Φ f (z) = dζ = Φ dζ = Φ F (z) 2πi ζ −z 2πi ζ −z |ζ−z0 |=r

|ζ−z0 |=r

for all Φ ∈ E . By the Hahn-Banach theorem this implies that f (z) = F (z).

We conclude this section with some applications of Theorem 1.6.1. 1.6.2 Theorem. Let D ⊆ C be a simply connected open set (Section 1.4.3), let E be a Banach space, and let f : D → E be holomorphic. Then there exists a holomorphic function F : D → E such that F = f . Proof. Fix z0 ∈ D. Then, for each z ∈ D, we choose a C 1 -function γ : [0, 1] → D with γ(0) = z0 and γ(1) = z and define 1 F (z) =

f γ(ζ) γ (ζ)dζ.

0

As D is connected, such a function γ always exists, and, as D is even simply connected, by the homotopy version of the Cauchy integral Theorem 1.4.4, this definition does not depend on the choice of γ. So F is well defined. It remains to prove that F is holomorphic and F = f . Let E be the dual of E. Then, by the corresponding scalar fact, for each Φ ∈ E , the function Φ ◦ F is holomorphic and (Φ ◦ F ) = Φ ◦ f . By the HahnBanach criterion, Theorem 1.6.1, this means that F itself is holomorphic, and thus this implies that F = f . 1.6.3 Lemma. Let D ⊆ C be an open set, let E be a Banach space, and let f : D → E be a function. Let E be the dual of E, and suppose there exists a sequence

1.6. The Hahn-Banach criterion

17

of holomorphic functions fn : D → E, n ∈ N, such that, for all Φ ∈ E and each compact K ⊆ D,

(1.6.2) lim max Φ f (z) − Φ fn (z) = 0. n→∞ z∈K

Then f is holomorphic on D. Proof. From (1.6.2) it follows that Φ ◦ f is holomorphic for each Φ ∈ E . Hence, by Theorem 1.6.1, f is holomorphic. 1.6.4 Theorem. Let D ⊆ C be an open set, let E be a Banach space, and let fn : D → E, n ∈ N, be a sequence of holomorphic functions, which converges, uniformly on each compact subset of D, to some function f : D → E. Then f is holomorphic on D, and the sequence of complex derivatives fn converges, uniformly on each compact subset of D, to f . Proof. It follows immediately from Lemma 1.6.3 that f is holomorphic. It remains to prove that fn converges to f , uniformly on each compact subset of D. Let z0 ∈ D and r > 0 be given such that the closed disc |z − z0 | ≤ r is contained in D. It is sufficient to prove that lim

max f (z) − fn (z) = 0.

(1.6.3)

n→∞ |z−z0 |≤r

Choose r with r < r such that also the closed disc |z − z0 | ≤ r is contained in D. Then, by the Cauchy formula for the complex derivative (1.5.6), f (ζ) − fn (ζ) 1 f (z) − fn (z) = dζ 2πi (ζ − z)2 |ζ−z0 |=r

for |z − z0 | ≤ r. By the estimate from Proposition 1.3.6 this implies that

f (z) − fn (z) ≤ r

max

|ζ−z0 |=r

f (ζ) − fn (ζ) r ≤ |ζ − z|2 (r − r)2

max

|ζ−z0 |=r

f (ζ) − fn (ζ)

for |z − z0 | ≤ r. As fn converges to f , uniformly on |z − z0 | = r , this implies 15.3.08. We already observed that also the holomorphic functions with values in a Banach space satisfy the Cauchy-Riemann equation (1.1.1). If we additionally assume that the function is continuously differentiable (with respect to the canonical real coordinates), then, as in the scalar case, the Cauchy-Riemann equation is also sufficient for holomorphy: 1.6.5 Theorem (Cauchy-Riemann criterion). Let D ⊆ C be an open set, let E be a Banach space, and let f : D → E be a C 1 -function. Then f is holomorphic, if and only if, ∂f ∂f =i on D. (1.6.4) ∂x ∂y

18


Proof. We already observed that (1.6.4) is necessary for the holomorphy of f . Now we assume that (1.6.4) is satisfied. Let E be the dual of E. Since f is of class C 1 , then it follows from (1.6.4) that, for each Φ ∈ E , ∂(Φ ◦ f ) ∂(Φ ◦ f ) =i ∂x ∂y

on D.

Hence, Φ ◦ f is holomorphic for all Φ ∈ E , and it follows from the Hahn-Banach criterion, Theorem 1.6.1, that f is holomorphic.

1.7

A criterion for the holomorphy of operator functions

1.7.1 Theorem. Let D ⊆ C be an open set, let E, F be Banach spaces, let A : D → L(E, F ) be a holomorphic operator function. Then the following two conditions are equivalent: (i) A is holomorphic on D. (ii) For each vector x ∈ E, the vector function Ax is holomorphic on D. Proof. (i) ⇒ (ii) is obvious. Assume (ii) is satisfied. We first prove that then A is continuous. Consider an arbitrary point z0 ∈ D and choose r > 0 so small that the disc |z − z0 | ≤ r is contained in D. Since, for each x ∈ E, Ax is holomorphic on D and, hence, Ax is bounded on |z − z0 | ≤ r, the set A(z)x |z − z0 | ≤ r is bounded for each x ∈ E. By the Banach-Steinhaus theorem, it follows that the set A(z) |z − z0 | ≤ r is bounded in L(E, F ), i.e., we have a constant C < ∞ such that

A(z) ≤ C

for all |z − z0 | ≤ r.

(1.7.1)

Now let zn , n ∈ N∗ , be a sequence which converges to z0 such that |zn − z0 | < r for all n ∈ N∗ . From the Cauchy formula, the estimate from Proposition 1.3.6 and estimate (1.7.1) then it follows that, for all x ∈ E, A(z0 )x − A(zn )x = 1 2π

A(z)x |z−z0 |=r

1 1 − z − z0 z − zn

dz

1 1 . ≤ rC x max − z − zn |z−z0 |=r z − z0

1.8. Power series Hence

19

1 1 − ,

A(z0 ) − A(zn ) ≤ rC max z − zn |z−z0 |=r z − z0

which further implies that limn→∞ A(zn ) = A(z0 ). To prove that A is holomorphic, we again consider an arbitrary point z0 ∈ D and choose r > 0 so small that the disc |z − z0 | ≤ r is contained in D. As A is continuous, by Lemma 1.5.2 the function 1 A(ζ) F (z) := dζ , |z − z0 | < r, 2πi ζ −z |ζ−z0 |=r

is holomorphic. Therefore it remains to prove that A(z) = F (z) for |z − z0 | < r. Let such z be given. Since, for each x ∈ E, the function Ax is holomorphic, it follows from the Cauchy formula and then from the definition of F that 1 A(ζ)x A(ζ) 1 A(z)x = dζ = dζ x = F (z)x 2πi ζ −z 2πi ζ −z |ζ−z0 |=r

|ζ−z0 |=r

for all x ∈ E. Hence A(z) = F (z).

1.8

Power series

1.8.1. Let E be a Banach space, and let ∞

an (z − z0 )n ,

z0 ∈ C,

(1.8.1)

n=0

be a power series with coefficients an ∈ E. The series (1.8.1) is called convergent in a point ζ ∈ C if the series of vectors ∞

an (ζ − z0 )n

n=0

converges in E (with respect to the norm), and it is called absolutely convergent in ζ ∈ C if ∞

an |ζ − z0 |n < ∞. n=0

1.8.2 Theorem (Abel’s lemma). Let E be a Banach space, and let ∞ n=0

(z − z0 )n an ,

z0 ∈ C,

(1.8.2)

20


be a power series with coefficients an ∈ E. Set ρ=

1 . lim sup n an

(1.8.3)

n→∞

If |ζ − z0 | > ρ, then the series (1.8.2) does not converge in ζ. If |ζ − z0 | < ρ, then the series (1.8.2) converges absolutely in ζ. Moreover, then, for all r < ρ, ∞ n=0

max |ζ − ζ0 |n an < ∞.

|ζ−z0 |≤r

(1.8.4)

Proof. First let |ζ − z0 | > ρ. This means that |ζ − z0 | lim sup an 1/n > 1, n→∞

i.e., lim sup an |ζ − z0 |n > 1. n→∞

Hence the sequence an (ζ − z0 )n does not converge to zero. Now let r < ρ be given. Then r lim sup an 1/n < 1. n→∞

Choose q with r lim sup n→∞

n

an < q < 1.

Then we can find n0 ∈ N such that r n an ≤ q It follows that

∞ n=0

for all n ≥ n0 .

max an |ζ − ζ0 |n ≤

|ζ−z0 |≤r

∞

q n < ∞.

n=0

1.8.3. The number ρ ∈ [0, ∞] defined by (1.8.3) is called the radius of convergence of the power series in (1.8.1). 1.8.4 Theorem. Let E be a Banach space, let ∞

(z − z0 )n an ,

z0 ∈ C,

(1.8.5)

n=0

be a power series with coefficients an ∈ E, and let ρ be the radius of convergence of it. Then:

1.8. Power series

21

(i) The power series

∞

n(z − z0 )n−1 an

(1.8.6)

n=1

also has the radius of convergence ρ. (ii) The function defined by f (ζ) =

∞

(ζ − z0 )n an ,

|ζ − z0 | < ρ,

(1.8.7)

n=0

is holomorphic on the open disc |z − z0 | < ρ, and f (ζ) =

∞

n(ζ − z0 )n−1 an ,

|ζ − z0 | < ρ.

(1.8.8)

n=1

(iii) If f is the holomorphic function defined by (1.8.6), then an =

f (n) (z0 ) , n!

n ∈ N.

(1.8.9)

Proof. Part (i): By definition (1.8.3) of the radius of convergence, assertion (i) is equivalent to the equality 1 n

lim sup n→∞

an

=

1 lim sup

. n an

n−1

n→∞

But the latter relation follows (for example) from the fact that the scalar power series ∞ ∞

an (z − z0 )n and n an (z − z0 )n−1 n=0

n=1

have the same radius of convergence. Part (ii): From (1.8.4) it follows that the sequence of partial sums N

an (z − z0 )n ,

N ∈ N,

n=0

converges to f , uniformly on each compact subset of the disc |z − z0 | < ρ. By Theorem 1.6.4 this implies that f is holomorphic and that the sequence of partial sums N nan (z − z0 )n−1 , N ∈ N, n=1

converge to f , uniformly on each compact subset of the disc |z − z0 | < ρ. Part (iii): This follows, as in the scalar case, by repeated application of (i) and (ii).

22


1.8.5 Theorem. Let D ⊆ C be an open set, let E be a Banach space, and let f : D → E be a holomorphic function. Let z0 ∈ D and let r > 0 such that the open disc |z − z0 | < r is contained in D. Then there exists a uniquely determined power series ∞ (z − z0 )n fn n=0

with coefficients fn ∈ E with radius of convergence such that f (ζ) =

∞

(ζ − z0 )n fn

(1.8.10)

n=0

for all ζ in a neighborhood of z0 . Then

f (n) (z0 ) 1 fn = = n! 2πi

|z−z0

|=r

f (z) dz (z − z0 )n+1

for all 0 < r < r.

(1.8.11)

Moreover, then ρ ≥ r and (1.8.10) holds for all ζ wtih |ζ − z0 | < r. Proof. The statement on uniqueness and the first equality in (1.8.10) follows from part (iii) of Theorem 1.8.4. To prove the remaining statements, we define vectors by f (z) 1 fn = dz , n ∈ N, (1.8.12) 2πi (z − z0 )n+1 |z−z0 |=r

where 0 < r < r. (By the Cauchy integral theorem this is independent of the choice of r .) It remains to prove that, for |ζ − z0 | < r, the series ∞

(ζ − z0 )n fn

n0

converges to f (ζ). Let such ζ be given. Then, as in the scalar case, we choose r with |ζ − z0 | < r < r and obtain by means of the Cauchy formula 1 f (z) 1 f (z) 1 f (ζ) = dz dz = ζ−z0 z − z 2πi z−ζ 2πi 1 − z−z 0 |z−z0 |=r

=

1 2πi

|z−z0 |=r

=

∞ n=0

(ζ − z0 )n

∞ n=0

1 2πi

ζ − z0 z − z0

n

|z−z0 |=r

0

|z−z0 |=r

f (z) dz z − z0

f (z) dz (z − z0 )n+1

=

∞ n=0

(ζ − z0 )n fn .

1.9. Laurent series

1.9

23

Laurent series

1.9.1 Theorem. Let z0 ∈ C, let 0 ≤ r < R ≤ ∞, let E be a Banach space, and let f be an E-valued function defined and holomorphic in r < |z − z0 | < R. Then there exists a uniquely determined Laurent series ∞

(z − z0 )n fn

(1.9.1)

n=−∞

with coefficients fn ∈ E such that ∞ n=−∞

max

r ≤|ζ−z0 |≤R

and

∞

f (ζ) =

for r < r < R < R

|ζ − z0 |n fn < ∞

(ζ − z0 )n fn

for r < |ζ − z0 | < R.

(1.9.2)

(1.9.3)

n=−∞

Moreover, then fn =

1 2πi

|z−z0 |=ρ

f (z) dz (z − z0 )n+1

for r < ρ < R.

(1.9.4)

Proof. Uniqueness and formula (1.9.4): Suppose we have a Laurent series with (1.9.2) and (1.9.3). Then, for each r < ρ < R, 1 2πi

|z−z0 |=ρ

∞ f (z) fk dz = (z − z0 )n+1 2πi k=−∞

|z−z0 |=ρ

(z − z0 )k dz = fn . (z − z0 )n+1

Existence: We define a function f+ on the disc |z − z0 | < R as follows: If a point z with |z −z0 | < R is given, then we choose a number ρ with |z −z0 | < ρ < R and set f (ζ) 1 f+ (z) = dζ. (1.9.5) 2πi ζ −z |ζ−z0 |=ρ

By the Cauchy integral theorem, this definition is independent of the choice of ρ, and, by Lemma 1.5.2, f+ is holomorphic on the disc |z − z0 | < R. Furthermore, we define a function f− on |z − z0 | > r as follows: If a point z with |z − z0 | > r is given, then we choose a number ρ with |z − z0 | > ρ > r and set f (ζ) 1 dζ. (1.9.6) f− (z) = − 2πi ζ −z |ζ−z0 |=ρ

24


Again, by the Cauchy integral theorem, this definition is independent of the choice of ρ, and, by Lemma 1.5.2, f− is holomorphic on |z − z0 | > r and lim f− (z) = 0.

(1.9.7)

|z|→∞

Then for r < |z − z0 | < R.

f (z) = f+ (z) + f− (z)

(1.9.8)

Indeed, let z with r < |z − z0 | < R be given. Then we choose ρ+ and ρ− with r < ρ− < |z − z0 | < ρ+ < R and obtain, by the Cauchy-Integral formula, 1 f (ζ) f (ζ) 1 f (z) = dζ − dζ = f+ (z) + f− (z). 2πi (ζ − z) 2πi (ζ − z) |ζ−z0 |=ρ+

|ζ−z0 |=ρ−

As f+ is holomorphic on the disc |z − z0 | < R, it can be represented by a power series (Theorem 1.8.5): f+ (z) =

∞

(z − z0 )n fn ,

|z − z0 | < R,

(1.9.9)

n=0

where, by Abel’s lemma (Theorem 1.8.2), ∞ n=0

max

|z−z0 |≤R

for R < R.

|z − z0 |n fn < ∞

(1.9.10)

Now we consider the function

1 , F (z) := f− z0 + z

which is defined and holomorphic for 0 < |z| < 1/r. As lim|z|→∞ f− (z) = 0, this function extends continuously to 0 with F (0) := 0. By Theorem 1.5.4, this extended F is holomorphic on the whole disc |z| < 1/r. Therefore also F can be represented by a power series F (z) =

∞

|z|
r.

max |z|n Fn < ∞ ,

|z|≤1/r

(1.9.11)

(1.9.12)

Set fn = F−n for n ≤ −1. Then it follows from (1.9.11) and (1.9.12) that f− (z) = F

1 z − z0

=

∞ n=1

1 z − z0

n Fn =

−1

(z − z0 )n fn ,

n=−∞

|z − z0 | > r,

1.10. Isolated singularities

25

and −1 n=−∞

max

|z−z0 |≥r

|z − z0 | fn = n

∞ n=1

1 n Fn < ∞ , max 1/|z−z0 |≤1/r z − z0

r > r.

Together with (1.9.8), (1.9.9) and (1.9.10) this shows that the Laurent series ∞

(z − z0 )n fn

n=−∞

has the required properties.

1.10

Isolated singularities

1.10.1. Let D ⊆ C be an open set, let E be a Banach space, and let f be an E-valued holomorphic function with the domain of definition D. Then, as in the scalar case, a point z0 ∈ C is called an isolated singularity of f , if {z0 } ∪ D is open and z0 ∈ D. 1.10.2. Let z0 ∈ C, let U be a neighborhood of z0 , let E be a Banach space, and let f : U \ {z0 } → E be a holomorphic function. (In other words, in the sense of Section 1.10.1, we assume that z0 is an isolated singularity of some holomorphic function f .) Further let ε > 0 be the maximal radius such that the punctured disc 0 < |z − z0 | < ε is still contained in U . By Theorem 1.9.1 then there exists a uniquely determined Laurent series ∞

(z − z0 )n fn

n=−∞

which converges in 0 < |z − z0 | < ε such that f (ζ) =

∞

(ζ − z0 )n fn

for all 0 < |ζ − z0 | < ε.

(1.10.1)

n=−∞

This Laurent series will be called the Laurent series of f at z0 , the formula (1.10.1) will be called the Laurent expansion of f at z0 , and the vector f−1 will be called the residuum of f at z0 . As in the scalar case one sees that, for each ε > 0 such that the punctured disc 0 < |z − z0 | ≤ ε is contained in U , 1 f−1 = f (z)dz. (1.10.2) 2πi |z−z0 |=ε

The isolated singularity z0 will be called a removable singularity of f if fn = 0 for all negative integers n. If, moreover, f0 = 0, then it will be called a zero of f .

26


If z0 is a zero of f and f ≡ 0 in a neighborhood of z0 , then (by uniqueness of the Laurent expansion) there exist positive integers n with fn = 0 – the smallest of them will be called the order of the zero z0 . The isolated singularity z0 will be called a pole of f if there exists a negative integer p such that fp = 0 and fn = 0 for all integers n ≤ p − 1. The integer p then is called the order of the pole z0 . If z0 is not a removable singularity of F and not a pole of f , then z0 is called an essential singularity of f . 1.10.3 Theorem (Riemann’s theorem on removable singularities). Let E be a Banach space, and let z0 be an isolated singularity of an E-valued holomorphic function f defined in a deleted neighborhood of z0 . If f is bounded, then z0 is removable as a singularity of f . Proof. Let f (z) =

∞

fn (z − z0 )n

n=−∞

be the Laurent series of f at z0 . We have to prove that fn = 0 for n ≤ −1. Let n ≤ −1 be given. Choose r > 0 sufficiently small. Then C :=

sup 0 0, c|z − z0 |N ≤ f (z) ≤ C|z − z0 |N

for 0 < |z − z0 | < ε.

(1.10.3)

(ii) The Laurent series of f at z0 is of the form f (z) =

∞ n=N

fn (z − z0 )n

with fN = 0.

(1.10.4)

1.10. Isolated singularities

27

Proof. First assume that (i) is satisfied. Then g(z) :=

f (z) (z − z0 )N

is a holomorphic function in a deleted neighborhood of z0 which is bounded from above and below. This implies, by Riemann’s Theorem 1.10.3, that g extends holomorphically to z0 , where g(z0 ) = 0. Let g(z) =

∞

(z − z0 )n gn

n=0

be the potential series of g at z0 . Then f (z) = (z − z0 )N g(z) =

∞

(z − z0 )n+N gn =

n=0

∞

(z − z0 )n gn−N

n=N

is the Laurent expansion of f . As g0 = g(z0 ) = 0, it is of the form (1.10.4). Now we assume that condition (ii) is satisfied. Then g(z) :=

∞ ∞ f (z) n−N = (z − z ) f = (z − z0 )n fn+N 0 n (z − z0 )N n=0 n=N

is holomorphic in a neighborhood of z0 , where g(z0 ) = fN = 0. Choose ε > 0 so small that

fN for z − z0 < ε. ≤ g(z) ≤ 2 fN 2 As f (z) = (z − z0 )N g(z), then (1.10.4) holds with C = 2|fN | and c = |fN |/2. 1.10.5 Theorem (Residue theorem). Let D ⊆ C be an open set with piecewise C 1 -boundary ∂D, let z1 , . . . , zn be a finite number of points in D, let E be a Banach space, and let f : D \ {z1 , . . . , zn } → E be a continuous function which is holomorphic in D. If we denote by reszj f the residuum of f at zj , then n

reszj f =

j=1

1 2πi

f (z)dz.

(1.10.5)

∂D

Proof. Choose ε > 0 so small that the closed discs |z − zj | ≤ ε, 1 ≤ j ≤ n, are pairwise disjoint and contained in D. Then by the Cauchy integral theorem f (z)dz = ∂D

By (1.10.2) this implies (1.10.5).

n

f (z)dz.

j=1 |z−zj |=ε

28


1.10.6. Let D ⊆ C be an open set, and let E be a Banach space. If we say that f is an E-valued holomorphic function with isolated singularities on D, then we mean that there is a set Z ⊆ D, which is discrete and closed in D (i.e., without accumulation points in D), such that f is an E-valued holomorphic function on D \ Z. In this case we say also that f : D → E is holomorphic with isolated singularities. The points from Z (which then are isolated singularities of f in the sense of Section 1.10.1) are called the singular points of f , and the points from D \ Z are called the regular points of f . If all singular points of f are either removable or poles, then f is called meromorphic on D.

1.11

Comments

Except for Theorem 1.2.2 about the strong maximum principle in Hilbert spaces (which is possibly new), the results of this chapter are coa.

Chapter 2

Solution of ∂u = f and applications In complex analysis of several variables, the inhomogeneous Cauchy-Riemann equation is an important tool. For results in Complex analysis of one variable this equation is also important, but it is missing in many standard books. For the aim of the present book, the inhomogeneous Cauchy-Riemann equation is basic. Therefore we dedicate the present chapter to it and its applications. We give these results with full proofs, not using the corresponding scalar fact, even if it would be possible to deduce a result by the Hahn-Banach theorem from the corresponding scalar fact. Moreover, here we present also some results, which are specific for scalar functions, with full proofs, as these results are difficult to find in the literature.

2.1

The Pompeiju formula for solutions of ∂u = f on compact sets

In this section, E is a Banach space. Let D ⊆ C be an open set, and let u : D → E be a C 1 -function. Then we use the abbreviation ∂u 1 ∂u ∂u ∂u = = +i ∂z 2 ∂x ∂y where x, y are the canonical real coordinates on C and z = x + iy. The function ∂u is called the Cauchy-Riemann derivative of u. By the Cauchy-Riemann criterion (Theorem 1.6.5), u is holomorphic if and only if ∂u = 0. This implies that, for all holomorphic functions h : D → C, ∂(hu) = h∂u.

(2.1.1)

30

Chapter 2. Solution of ∂u = f and applications

Moreover, if ϕ : D → C is a C 1 -function with compact support, then partial integration gives ϕ(∂u) dλ = − (∂ϕ)u dλ D

D

where dλ is the Lebesgue measure. This can be used to define ∂u in the sense of distributions if u is not of class C 1 . In this book, we are interested only in the following special case: 2.1.1 Definition. Let D ⊆ C be an open set, and let u : D → E be a continuous function. We say that u has a continuous Cauchy-Riemann derivative if there exists a continuous function v : U → E such that ϕv dλ = − (∂ϕ)u dλ (2.1.2) D

D

∞

for all C -functions ϕ : D → C with compact support. It is clear that this function v then is uniquely determined. We call it the Cauchy-Riemann derivative of u and denote it by ∂u. Instead of “u has a continuous Cauchy-Riemann derivative” we say also “∂u is continuous”. Moreover, if u and f are two continuous functions, then writing “∂u = f ” we mean that “u has a continuous Cauchy-Riemann derivative and ∂u = f ”. 2.1.2 Proposition. Let D ⊆ C be an open set, let u : D → E be a continuous function such that ∂u is continuous on D, and let ψ : D → C be a C 1 -function. Then also ∂(ψu) is continuous on D, and ∂(ψu) = (∂ψ)u + ψ∂u. Proof. For all C ∞ -functions ϕ : D → C with compact support, we have ϕ (∂ψ)u + ψ∂u dλ = ϕ(∂ψ)u dλ + ϕψ∂u dλ D

D ϕ(∂ψ)u dλ − ∂(ϕψ)u dλ

= D

D

ϕ(∂ψ)u dλ −

= D

=−

(∂ϕ)ψu dλ −

D

D

ϕ(∂ψ)u dλ D

(∂ϕ)ψu dλ. D

2.1.3 Lemma. Let D ⊆ C be an open set, let u : D → E be a continuous function such that also ∂u is continuous on D, and let K ⊆ D be compact. Then there

2.1. The Pompeiju formula

31

exists a sequence (un )n∈N of C ∞ -functions un : D → E such that, uniformly on K, both lim ∂un = ∂u. lim un = u and n→∞

n→∞

Proof. Choose an open neighborhood V of K which is relatively compact in D. Further, let ϕ be a real non-negative C ∞ -function on C with ϕ dλ = 1 and ϕ(z) = 0 if |z| > 1. C

Now let a continuous map u : D → E be given such that ∂u is also continuous on D. Then we set z−ζ u(ζ) dλ(ζ), z ∈ C , ε > 0. (2.1.3) uε (z) = ε2 ϕ ε V Since z−ζ 2 ε dλ(ζ) = 1 for all z ∈ C ϕ ε C

z−ζ and ϕ ε

=0

if |z − ζ| > ε,

then, for sufficiently small ε > 0 and all z ∈ K, z − ζ uε (z) − u(z) ≤ ε2 u(ζ) − u(z). u(ζ) − u(z) dλ(ζ) ≤ max ϕ ε ζ∈V ,|z−ζ|≤ε V Since V is compact, it follows that limε→0 uε = u, uniformly on K. Since also ∂u is continuous on D, then, in the same way, we get limε→0 (∂u)ε = ∂u, uniformly on K. This completes the proof, because ∂uε = (∂u)ε . Indeed, differentiating under the integral in (2.1.3) we obtain z−ζ z−ζ 2 2 u(ζ) dλ(ζ) = −ε u(ζ) dλ(ζ), (∂uε )(z) = ε ∂z ϕ ∂ζ ϕ ε ε V V which implies, by (2.1.2), (∂uε )(z) = ε

2 V

ζ −z (∂u)(ζ) dλ(ζ) = (∂u)ε (z). ϕ ε

2.1.4 Theorem (Cauchy formula for continuous functions). Let D be a bounded open set with piecewise C 1 -boundary ∂D, and let u : D → E be a continuous function such that also ∂u is continuous on D. Moreover we assume that ∂u admits a continuous extension to D. Then u(ζ) ∂u(z) 1 1 u(z) = dζ − dλ(ζ), z ∈ D. (2.1.4) 2πi ζ −z π ζ −z ∂D

D

32

Chapter 2. Solution of ∂u = f and applications

Proof. First consider the case when u is of class C 1 in a neighborhood of D. Let z ∈ D. Set Dε = {ζ ∈ D | |ζ − z| > ε} for sufficiently small ε > 0. Then, by Stokes’ formula, with ζ = x + iy, ∂Dε

u(ζ) dζ = ζ −z

∂Dε

=−

Dε

Since lim

ε 0 ∂Dε

u(ζ) dx + i ζ −z

∂Dε

∂ u(ζ) dλ(ζ) + i ∂y ζ − z

u(ζ) dζ = ζ −z

u(ζ) dy ζ −z

Dε

∂D

u(ζ) dζ − lim ε 0 ζ −z

∂ u(ζ) dλ(ζ) = +2i ∂x ζ − z

Dε

|ζ−z|=ε

u(ζ) dζ = ζ −z

∂D

(δu)(ζ) dλ(ζ). ζ −z

u(ζ) dζ − 2πi u(z), ζ −z

this implies (2.1.4). Now we consider the case when u admits a continuous extension u to a neighborhood of D such that ∂ u is also continuous in this neighborhood. Then by Lemma 2.1.3 we can find a sequence (un )n∈N of C ∞ -maps un : C → E such that, uniformly on D, both lim un = u

n→∞

and

lim ∂un = ∂u.

n→∞

Since the required equation(2.1.4) is already proved for each un , passing to the limit, it follows for u. Finally, consider the general case. Then we take a sequence of bounded open sets (Dn )n∈N with piecewise C 1 -boundary such that Dn ⊆ D and u(ζ) u(ζ) ∂u(ζ) ∂u(ζ) lim dζ = dζ , lim dλ(ζ) = dλ(ζ). n→∞ n→∞ ζ −z ζ −z ζ −z ζ −z ∂Dn

Dn

∂D

D

Since the required equation(2.1.4) is already proved for each uD , passing to the n limit, it follows for u. 2.1.5 Theorem. Let D ⊆ C be an open set, and let u : D → E be a continuous function such that ∂u = 0 on D. Then u is holomorphic. Proof. Since the assertion is local, we may assume that D is a disc and u is defined and continuous in a neighborhood of D and that ∂u = 0 in this neighborhood. Then the second integral in the Cauchy formula (2.1.4) vanishes, i.e., 1 u(ζ) u(z) = dζ , z ∈ D. 2πi ζ −z ∂D

Now the assertion follows by differentiation under the integral.


33

2.1.6 Definition. Let 0 < α < 1. First let M ⊆ C be an arbitrary set. For any function f : M → E, we set

f M,0 = sup f (z) z∈M

and

f M,α = f M,0 +

sup z,w∈M,z =w

f (z) − f (w) . |z − w|α

We write f 0 instead of f M,0 and f α instead of f M,α if it is clear which set M we mean. A function f : M → E is called (locally) Hölder continuous with exponent α if, for each point z0 ∈ M , there exists a neighborhood U of z0 such that

f U ∩M,α < ∞ . The space of all such functions will be denoted by (C α )E (M ). Instead of H¨ older continuous with exponent α we say also Hölder-α continuous or of class C α . A function f : M → E will be called Hölder continuous if there exists 0 < α < 1 such that f is Hölder continuous with exponent α. Sometimes, for practical reasons, continuous functions will be called Hölder continuous functions with exponent 0 (although they are not H¨ older continuous). Now let D ⊆ C be an open set, and k ∈ N∗ ∪ {∞}. Recall that (C k )E (D) denotes the space of all E-valued C k functions on D. A function f : D → E is called of class C k+α or simply C k+α if f is C k on D and, moreover, the partial derivatives of order k of f are of class C α on D. The space of all such functions will be denoted by (C k+α )E (D). 2.1.7 Definition (Pompeiju operator ΠD ). Let D ⊆ C be a bounded open set. Then, for any bounded continuous function f : D → E and each z ∈ C, we define (ΠD f )(z) = −

1 π

D

f (ζ) dλ(ζ) . ζ −z

The operator ΠD will be called the Pompeiju operator on D. 2.1.8 Lemma. Let d < ∞. Then there exists C < ∞ such that 1 1 ζ − z − ζ − w dλ(ζ) ≤ C|z − w| log |z − w| |ζ| 12 |z − w|. Therefore dλ(ζ) π 2 dλ(ζ) = |z − w| . ≤ |ζ − w| |z − w| 2 |ζ−z|< 12 |z−w|

|ζ−z|< 12 |z−w|

Together with (2.1.5) this yields 1 3π 1 dλ(ζ) ≤ − |z − w| . ζ − z ζ − w 2 |ζ−z|< 12 |z−w|

Hence

min(|ζ−z|,|ζ−w|)< 12 |z−w|

It remains to estimate

1 1 ζ − z − ζ − w dλ(ζ) ≤ 3π|z − w| . 1 1 ζ − z − ζ − w dλ(ζ) .

I(z, w) := |ζ| 12 |z−w|

If |ζ − w| ≥ 12 |z − w|, then |ζ − z| ≤ |ζ − w| + |z − w| ≤ |ζ − w| + 2|ζ − w| = 3|ζ − w| and therefore

1 |z − w| 1 1 |z − w| ζ − z − ζ − w = |ζ − z||ζ − w| ≤ 3 |ζ − z|2 .

Hence I(z, w) ≤

|z − w| 3

dλ(ζ) |z − w| ≤ 2 |ζ − z| 3

|ζ| 12 |z−w|

|z − w| = 3

2d 1 2 |z−w|

|ζ−z|=r

2d>|ζ−z|> 12 |z−w|

|dζ| 2π dr = |z − w| 2 r 3

2d 1 2 |z−w|

dr . r

dλ(ζ) |ζ − z|2


35

Since 2d

1

dr = log(2d) − log |z − w| = log(4d) − log |z − w| r 2

1 2 |z−w|

≤ log(4d) + log |z − w|,

this yields I(z, w) ≤

2π log(4d) 2π |z − w| + |z − w| log |z − w|. 3 3

Together with (2.1.7) this implies that there is a constant C < ∞ (depending on d) satisfying (2.1.5) 2.1.9 Theorem. Let D ⊆ C be a bounded open set, let ΠD be the Pompeiju operator (Def. 2.1.7), and let f : D → E be a bounded continuous function. (i) Then there exists a constant C < ∞ such that (ΠD f )(z) − (ΠD f )(w) ≤ C|z − w| log |z − w| for all z, w ∈ C. In particular, ΠD f ∈ D

(C α )E (D) .

(2.1.9)

0 R ⊆ D− , and let f− (z) =

∞

fn z n

n=−∞

be the Laurent expansion of f− with respect to z ∈ C |z| > R . Set h(z) =

∞

fn z n

for z ∈ C

and f− (z) =

fn z n

for |z| > R .

n=−∞

n=0

Then

−1

f− (z) = f− (z) − h(z)

for |z| > R.

As f− is continuous on D− and holomorphic in D− and h is holomorphic in C, this shows that f− admits a continuous extension to D− , which is holomorphic in D− . Moreover, by definition of f− , it is clear that f− extends holomorphically to ∞, where f− (∞) = 0. As f+ is continuous on D+ and holomoprhic in D+ , and h is holomorphic in C, in the same way we see that f+ := f+ − h is continuous on D+ and holomorphic in D+ . Since f− = h + f− , from (3.7.6) it follows that f = f+ − f− = f+ − h − f− = f+ − f−

on Γ .

3.8

Factorization of scalar functions with respect to a contour. First remarks

In this section, D+ ⊆ C is a bounded open set with piecewise C 1 -boundary Γ oriented by D+ (Section 1.4.1) such that 0 ∈ D+ , and D− = C \ D+ . We denote by Γ0 , . . . , Γm the connected components of Γ, endowed with the orientation of Γ, so that −Γ0 is the boundary of the unbounded component of D− . 3.8.1 Definition. Let f : Γ → C∗ be a continuous function. We say that f admits a factorization with respect to Γ if there exist continuous functions f− : D− ∪{∞} → C∗ and f+ : D+ → C∗ , which are holomorphic in D− ∪ {∞} (Section 3.1.1) and D+ , respectively, such that, for some integer κ, f (z) = z κ f+ (z)f− (z)

for z ∈ Γ .

(3.8.1)

To underline the difference with Definition 3.11.1 below, in this case we also say that f admits a global factorization with respect to Γ.

3.8. Factorization. First remarks

85

If f is a scalar rational function which is holomorphic and = 0 on Γ0 ∪. . .∪Γm , then a factorization of f with respect to Γ0 ∪ . . . ∪ Γm is easy to find. Indeed, then f is of the form p − p+ f= q− q+ where p− , q− , p+ , q+ are polynomials such that p− , q− have no zeros on D− and p+ , q+ have no zeros on D+ . Let deg p− and deg q− be the degrees of p− and q− , respectively, and set κ = deg p− − deg q− . Then the functions p− /q− and p+ /q+ are holomorphic on D− and D+ , respectively, p− (z) a := lim z −κ z→∞ q− (z) exists and a = 0. Therefore we obtain a factorization of f with respect to Γ0 ∪ . . . ∪ Γm , by setting p− (z) for z ∈ D− q− (z) p+ (z) f+ (z) = for z ∈ D+ . q+ (z)

f− (z) = z −κ

and f− (∞) = a,

Not every continuous function f : Γ0 ∪ . . . ∪ Γm → C∗ admits a factorization with respect to Γ0 ∪. . .∪Γm (see Remark 3.11.4 below). To study the factorization problem in the general case, now, also for continuous functions, we define the index, which was introduced for holomorphic functions already in Definition 2.5.3. For that let Γ be the union of some of the connected components Γ0 , . . . , Γm of Γ, oriented in an arbitrary way. First consider two holomorphic functions f, g : Γ → C∗ such that f (z)g(z)−1 − 1 < 1 for all z ∈Γ . Then this estimate holds also in some neighborhood of Γ . Therefore log f g −1 is defined in this neighborhood, where log is the main branch of the logarithm, and

−1 −1 elog(f g ) fg = = f g −1 . −1 −1 fg fg Hence

indΓ f − indΓ g = indΓ f g

−1

1 = 2πi

f (z)g −1 (z) dz = 0 .

Γ

Therefore the following definition is correct (and agrees with Definition 2.5.3 when f is holomorphic):

86

Chapter 3. Splitting and factorization

3.8.2 Definition. Let Γ be the union of some of the connected components Γ0 , . . ., Γm of Γ, oriented in an arbitrary way. Let f : Γ → C∗ be a continuous function. Then we define indΓ f = indΓ f (3.8.2) where f : Γ → C∗ is a holomorphic function which is sufficiently close to f , uniformly on Γ . (By Corollary 3.3.3 such a function f exists.) This number indΓ f is called the index of f with respect to Γ . By Proposition 2.5.4 (iv), it is an integer. 3.8.3 Proposition. (i) Let Γ be the union of some of the connected components Γ0 , . . . , Γm of Γ, oriented in an arbitrary way. Let f, g : Γ → C∗ be two continuous functions. Then indΓ (f g) = indΓ f + indΓ g .

(3.8.3)

(ii) Let f : Γ → C∗ be a continuous function which admits an extension to D+ which is continuous on Γ and meromorphic in D+ . Denote this extension also by f , let N be the number of zeros of f in D+ , counted according to their multiplicities, and let P be the number of poles of f in D, also counted according to their multiplicities. Then (recall that Γ is oriented by D+ ) indΓ f = N − P .

(3.8.4)

(iii) Let f : Γ → C∗ be a continuous function which admits an extension to D− ∪ {∞} which is continuous on Γ and meromorphic in D− ∪ {∞} (Section 3.1.1). Denote this extension also by f , let N be the number of zeros of f in D− ∪ {∞}, counted according to their multiplicities, and let P be the number of poles of f in D− ∪ {∞}, also counted according to their multiplicities. Then (recall that Γ is oriented by D+ ) then − indΓ f = N − P .

(3.8.5)

Proof. By our definition of the index in the case of a continuous function, part (i) follows immediately from part (i) of Proposition 2.5.4. Consider part (ii). Using the Mergelyan approximation Theorem 2.2.1 and Cauchy’s theorem it is easy to see that D+ can be replaced by a slightly smaller set. Then the assertion follows from part (ii) of Proposition 2.5.4. In the same way part (iii) follows from part (iii) of Proposition 2.5.4. 3.8.4 Corollary. Let f : Γ→ C be a continuous function, and let f (z) = z κ f+ (z)f− (z) be a factorization of f with respect to Γ. Then the integer κ is uniquely determined, namely (recall that Γ is oriented by D+ ) κ = indΓ f.

3.8. Factorization. First remarks

87

Proof. Since 0 ∈ D+ , it follows from Proposition 3.8.3 that indΓ f = indΓ z κ + indΓ f+ + indΓ f− = indΓ z κ = κ.

3.8.5 Proposition. Let 0 ≤ j ≤ m, and let f : Γj → C∗ be a continuous function with indΓj f = 0. Then there is a continuous function log f : Γj → C with elog f = f

on Γj .

Note that log f is uniquely determined up to an additive constant of the form k2πi, k ∈ Z. Proof. Choose a holomorphic function f : Γj → C∗ which is so close to f that for z ∈ Γj , f (z)f−1 (z) − 1 < 1 and

indΓj f = indΓj f = 0 ,

0 ≤ j ≤ m.

By the first relation log(f f−1 ) is well defined on Γj , and, by Theorem 2.5.5, from the second relation it follows that there exists a holomorphic function h : Γ → C with eh = f. It remains to set log f = h + log(f f−1 ), where the latter log denotes the main branch of the complex logarithm. 3.8.6. Let f : Γ → C be a continuous function, and let f (z) = z κ f+ (z)f− (z) be a factorization of f with respect to Γ. By multiplying by a constant we can always achieve that f− (∞) = 1. With this additional property, the factorization of f with respect to Γ is uniquely determined. Indeed, let f = z κ f+ f− be a second factorization of f . Then −1 −1 f+ f+ = f− f−

on Γ .

Therefore (cf. Theorem 1.5.4) the function defined by on D+ , f f−1 h = + +−1 f− f− on D− ∪ {∞} , is a well-defined holomorphic function on C ∪ {∞} which is equal to 1 at ∞. Hence this function is identically equal to 1, i.e., f± = f± . 3.8.7 Theorem. If the contour Γ is not connected, i.e., m ≥ 1, then, for 1 ≤ j ≤ m, we denote by Uj the bounded connected component of D− with boundary −Γj and we fix some point pj ∈ Uj . Let f : Γ → C be a continuous function. Set κj := indΓj f ,

0 ≤ j ≤ m.

88


Further, let g : Γ → C be one of the continuous functions with for z ∈ Γ0 , z −κ0 f (z) , eg(z) = (z − pj )κj f (z) , for z ∈ Γj , 1 ≤ j ≤ m, (which exists by Proposition 3.8.5). Then the following are equivalent: (i) The function f admits a factorization with respect to Γ. (ii) The function g splits with respect to Γ. Proof. First assume that there is a factorization f (z) = f− (z)z κ f+ (z) of f . Then for 0 ≤ j ≤ m,

indΓj f− = 0 indΓ0 f+ = κ0 − κ, indΓj f+ = κj

for 1 ≤ j ≤ m.

Therefore it follows from the Mergelyan approximation Theorem 2.2.1 and Theorem 2.5.5 that there exists a continuous function g− : D− ∪ {∞} → C, which is holomorphic in D− ∪ {∞}, such that f− = eg− on D− ∪ {∞}. Moreover, set for z ∈ Γ0 , z κ−κ0 f+ (z), f+ (z) = κ κj z (z − pj ) f+ (z), for z ∈ Γj , 1 ≤ j ≤ m. Then indΓj fj = 0 for 0 ≤ j ≤ m, and from Theorem 2.5.5 and the Mergelyan Theorem 2.2.1 we get a contiunous function g+ : D+ → C, which is holomorphic in D+ , such that f+ = eg+ on D+ . It follows eg(z) = z −κ0 f (z) = z −κ0 f− (z)z κ f+ (z) = f− (z)f+ (z) = eg− (z)+g+ (z) for z ∈ Γ0 , and eg(z) = (z − pj )κj f (z) = (z − pj )κj f− (z)z κ f+ (z) = f− (z)f+ (z) = eg− (z)+g+ (z) for z ∈ Γj , 1 ≤ j ≤ m. Hence, there are some integers μj with for z ∈ Γj ,

g(z) = g+ (z) + g− (z) + μj

0 ≤ j ≤ m.

Setting g− (z) = g− (z) + μj

for z ∈ Γj ,

0 ≤ j ≤ m,

we get a splitting (g+ , g− ) of g. Now we assume that a splitting g = g+ + g− of g is given. Then z −κ0 f (z) = eg(z) = eg− (z) eg+ (z) (z − pj ) f (z) = e κj

g(z)

g− (z) g+ (z)

=e

e

for z ∈ Γ0 , for z ∈ Γj , 1 ≤ j ≤ m.

3.9. Factorization of H¨ older functions

89

Therefore it is sufficient to prove that the function ϕ defined by if z ∈ Γ0 , z κ0 ϕ(z) = −κj (z − pj ) if z ∈ Γj , 1 ≤ j ≤ m, admits a factorization. Let U0 be the unbounded connected component of D− , and let Uj , 1 ≤ j ≤ m, be the bounded connected component of D− with the boundary −Γj , 1 ≤ j ≤ m. Set ϕ+ (z) = (z − p1 )−κ1 · . . . · (z − pm )−κm

for D+ ,

and for z ∈ D− we define z κ0 −κ (z − p1 )−κ1 · . . . · (z − pm )−κm ϕ− (z) = z −κ (z − p1 )κ1 · . . . !j . . . · (z − pj )κm

if z ∈ U 0 , if z ∈ U j , 1 ≤ j ≤ m.

Since pj ∈ U − j and 0 ∈ D+ , then φ+ is holomorphic and = 0 on D+ , ϕ− is holomorphic and = 0 on D− ∪ {∞} and ϕ(z) = z κ ϕ− (z)ϕ+ (z)

3.9

for z ∈ Γ.

Factorization of Hölder functions

Here we use the notations and definitions introduced in the preceding Section 3.8, and we prove: older continuous with 3.9.1 Theorem. Let 0 < α < 1, and let f : Γ → C∗ be H¨ exponent α (Def. 2.1.6). Then f admits a factorization with respect to Γ. older continuous If f = z κ f+ f− is an arbitrary factorization of f , then f+ is H¨ with exponent α on D+ , and f− is H¨ older continuous with exponent α on D− . If, moreover, Γ is of class C k and f is of class C k+α on Γ (Def. 3.4.1) for some k ∈ N∗ , and f = z κ f+ f− is an arbitrary factorization of f , then f+ is of class C k+α on D+ , and f− is of class C k+α on D− . Proof. Let k ∈ N and assume that if k = 0, then f is Hölder-α continuous, and if k ≥ 1, then Γ is of class C k and f is of class C k+α . As observed in Section 3.8.6, up to a multiplicative constant, the solution of the factorization problem is uniquely determined. Therefore it is sufficient to prove the existence of at least one factorization f = z κ f+ f− such that the factors f± are of class C k+α on D± . Set κj = indΓj f,

0 ≤ j ≤ m,

and

κ = κ0 + κ1 + . . . + κm .

90


Let U1 , . . . , Um be the bounded connected components of D− so that −Γj is the boundary of Uj . Choose points pj ∈ Uj and set f(z) = z −κ (z − p1 )κ1 · . . . · (z − pm )κm f (z). Since Γ0 is the boundary of the simply connected open set D+ ∪ U 1 ∪ . . . ∪ U m (Section 2.5.1) and 0, p1 , . . . , pm ∈ D+ ∪ U 1 ∪ . . . ∪ U m , it follows from Proposition 3.8.3 (ii) that

indΓ0 z −κ (z − p1 )κ1 · . . . · (z − pm )κm = −κ + κ1 + . . . + κm = −κ0 , In the same way we get

indΓj z −κ (z − p1 )κ1 · . . . · (z − pm )κm = −κj

for 1 ≤ j ≤ m.

Hence, by Proposition 3.8.3 (i), indΓj f = 0

for all 0 ≤ j ≤ m.

Therefore, by Proposition 3.8.5, we can find a continuous function g : Γ → C with eg = f. Since f is of class C k+α and, locally, the function g is of the form g = log f , where log is a branch of the logarithm (which is a holomorphic function), it follows that also g is of class C k+α . Therefore, by theorems 3.3.2 and (3.4.5), there exists a C k+α -function g+ : D+ → C, which is holomorphic in D+ , and a C k+α -function g− : D− ∪ {∞} → C, which is holomorphic in D− ∪ {∞}, such that g = g+ + g− on Γ. Then f = eg+ eg− on Γ. Hence, by definition of f, f=

z κ eg+ eg− (z − p1 )κ1 . . . (z − pm )κm

Setting f− = eg−

and f+ =

eg+ (z − p1 )κ1 . . . (z − pm )κm

we get a factorization f = z κ f+ f−

(3.9.1)

of f with respect to Γ. Since the functions g± are of class C k+α on D± , also the functions f± are of class C k+α on D± .

3.10. Factorization of Wiener functions

3.10

91

Factorization of Wiener functions

Here T is the unit circle, D+ is the open unit disc, and D− = C \ D+ . Denote by W (C) the space of functions f : Γ −→ C of the form ∞

f (z) =

fn z

n

with f W :=

n=−∞

∞

|fn | < ∞ .

(3.10.1)

n=−∞

It follows from Cauchy’s product theorem that if f, g ∈ W (C), then f g ∈ W (C) and

f g W ≤ f W g W . Hence, W (C) is a Banach algebra with the norm · W . Moreover: 3.10.1 Proposition. If f ∈ W (C) and f (z) = 0 for all z ∈ T, then f −1 ∈ W (T). Proof. Let f ∈ W (C) be given, which is not an invertible element of W (C). We have to find θ ∈ T with f (θ) = 0. Since f is not invertible as an element of W (A), by the theory of commutative Banach algebras, there exits a multiplicative functional Φ on W (C) with Φ(f ) = 0. For all fixed complex numbers λ ∈ C\T, the function z −λ is an invertible element of W (C) (as (z − λ)−1 is holomorphic in a neighborhood of T). Hence, for each fixed λ ∈ C \ T, Φ(z − λ) = 0 and, therefore, Φ(z) − λ = Φ(z − λ) = 0. ∞ Hence θ := Φ(z) ∈ T. Now let fn be the coefficients with f (z) = n=−∞ fn z n . Since |fn | < ∞, and, by definition, z n W = 1, then it follows that f (θ) =

∞ n=−∞

fn θn =

∞ n=−∞

∞ n fn Φ(z) = fn Φ(z n ) = Φ(f ) = 0.

n=−∞

3.10.2 Theorem. Let f ∈ W (C) such that f (z) = 0 for all z ∈ Γ. Then f admits a factorization f (z) = z κ f+ (z)f− (z) with respect to Γ such that f+ , f− ∈ W (C). Moreover, if f (z) = z κ f+ (z)f− (z) is a factorization of f with respect to Γ, then the factors f± automatically belong to W (C). Proof. As observed in Section 3.8.6, up to a multiplicative constant, the solution of the factorization problem is uniquely determined. Therefore it is sufficient to prove the existence of at least one factorization f = z κ f+ f− such that f+ , f− ∈ W (C). By Proposition 3.10.1, f −1 belongs to W (C). Since the functions, which are holomorphic on Γ, are dense in W (C), we can find a holomorphic function h on Γ with 1

h − f −1 W < .

f W

92


As the set f −1 (T) is a compact subset of C∗ and |f −1 (z) − h(z)| ≤ f −1 − h W for all z ∈ T, we can moreover achieve that h(z) = 0 for all z ∈ T. Set g = f h − 1. Then

g W = f (h − f −1 ) W ≤ f W h − f −1 W < 1

and

(3.10.2)

f = h−1 (1 + g).

From (3.10.2) it follows that the series ∞ (−1)n−1 n g n n=1

converges absolutely with respect to the norm · W to some element of W (A), which we denote by log(1 + g). Since convergence in W (A) implies pointwise convergence, it follows that log(1 + g) (z) = log 1 + g(z) for all z ∈ T, where the log on the right-hand side denotes the main branch of the complex logarithm. Since log(1 + g) is an element of W (A), it can be written also in the form ∞ ∞ log 1 + g(z) = z n an with

an W < ∞. n=−∞

Set v− (z) =

−1

z n an ,

n=−∞

v+ (z) =

n=−∞

∞

z n an ,

n=0

w+ = e and w− = e . So we get continuous functions w+ : D+ → C∗ , w− : D− ∪ {∞} → C∗ , which are holomorphic in D+ and D− ∪ {∞}, respectively, such that w+ , w− ∈ W (C), 1 + g = w− w+ and therefore v+

v−

f = h−1 w− w+

on Γ.

Since h is holomorphic on Γ, it follows from Theorem 3.9.1 that there exists a factorization h−1 (z) = z κ h− (z)h+ (z) (3.10.3) of h with respect to Γ. From (3.10.3) and Theorem 1.5.4 then we get that h+ and h− are holomorphic on Γ. Hence h+ and h− belong to W (C). Setting f± = w± h± , we get a factorization f (z) = z κ f− (z)f+ (z) of f with respect to Γ, where f+ , f− ∈ W (C).

3.11. The multiplicative local principle

3.11

93

The multiplicative local principle

In this section we use the notations and definition introduced in Section 3.8. In Section 3.7 we saw that there is a local principle for the splitting problem. Here we will show that there is a local principle also for the factorization problem (see Theorem 3.11.5 below). 3.11.1 Definition. Let U ⊆ C be an open set with U ∩Γ = ∅, and let f : Γ∩U → C∗ be a continuous function. We say that f admits a factorization over U with respect to Γ if there exist C∗ -valued functions f− and f+ , where f+ is continuous on U ∩D+ and holomorphic in U ∩ D+ , and f− is continuous on U ∩ D− and holomorphic in U ∩ D− , such that f = f− f+ on U ∩ Γ . (3.11.1) Then the pair (f− , f+ ) or the representation f = f− f+ will be called a factorization of f over U with respect to Γ. Let f : Γ → C∗ be a continuous function. We say that f admits local factorizations with respect to Γ if, for each w ∈ Γ, there exists a neighborhood U of w such that f admits a factorization over U with respect to Γ. 3.11.2 Lemma. Let f : Γ ∩ U → E be a continuous function, and let U ⊆ C be an open set with U ∩ Γ = ∅. If (f− , f+ ) and (f− , f+ ) are two factorizations of f over U with respect to Γ, then there is a (uniquely determined) holomorphic function h : U → C∗ with f− = hf−

on U ∩ D−

and

f+ = hf+

on U ∩ D+ .

(3.11.2)

Proof. By hypothesis f− f+ = f = f− f+ on U ∩ Γ. Then, setting # on U ∩ D+ , f+ f+ h := # f− f− on U ∩ D− , we get a continuous function h : U → C∗ , which satisfies (3.11.2) and which is holomorphic in U \ Γ. By Theorem 1.5.4, h is holomorphic on all of U . 3.11.3 Lemma. Let g : Γ → C be a continuous function. Then the following are equivalent: (i) The function eg admits local factorizations with respect to Γ. (ii) The function g locally splits (additively) with respect to Γ. Proof. (i)⇒(ii): Let w ∈ Γ be given. As eg admits local factorizations with respect to Γ, then there exist a neighborhood U of w and continuous functions u± : U ∩ D± → C∗ , which are holomorphic in U ∩ D± , such that eg = u+ u−

on Γ ∩ U.

(3.11.3)

94


After shrinking U , we may assume that a certain branch of the logarithm is defined on the values of u+ , and a certain other branch of the logarithm is defined on the values of u− , i.e., we have continuous functions v± : U ∩ D± → C, which are holomorphic in U ∩ D± , such that u± = ev± . Together with (3.11.4) and (3.11.3) this implies on Γ ∩ U. eg = ev+ +v− Hence, for some k ∈ Z (we may assume that U ∩ Γ is connected), g = (k2πi + v+ ) + v− on Γ ∩ U . (ii)⇒(i): Let w ∈ Γ be given. As g locally splits additively with respect to Γ, then there exist a neighborhood U of w and continuous functions v± : U ∩D± → C, which are holomorphic in U ∩ D± , such that on Γ ∩ U.

g = v+ + v−

Then eg = ev+ ev− .

3.11.4 Remark. Together with the example from Section 3.6 this lemma shows that not any continuous function f : Γ → C∗ admits local factorizations with respect to Γ. The fact stated by the following theorem will be called the multiplicative local principle: 3.11.5 Theorem. Let f : Γ → C∗ be a continuous function which admits local factorizations with respect to Γ. Then f admits a global factorization with respect to Γ. Proof. Recall that Γ = ∂D+ is oriented by D+ (by Definition 2.1.6) and that Γ0 , . . . , Γm are oriented in the same way. Set κj = indΓj f,

0 ≤ j ≤ m,

and

κ = κ0 + κ1 + . . . + κm .

Let U1 , . . . , Um be the bounded connected components of D− . Choose points pj ∈ Uj and set f(z) = z −κ (z − p1 )κ1 · . . . · (z − pm )κm f (z). Since Γ0 is the boundary of the simply connected open set D+ ∪ U 1 ∪ . . . ∪ U m (Section 2.5.1), the orientation included, and since 0, p1 , . . . , pm ∈ D+ ∪ U 1 ∪ . . . ∪ U m , it follows from Proposition 3.8.3 (ii) that

indΓ0 z −κ (z − p1 )κ1 · . . . · (z − pm )κm = −κ + κ1 + . . . + κm = −κ0 , In the same way we get

indΓj z −κ (z − p1 )κ1 · . . . · (z − pm )κm = −κj

for 1 ≤ j ≤ m.

3.12. Comments

95

Hence, by Proposition 3.8.3 (i), indΓj f = 0

for all 0 ≤ j ≤ m.

Therefore, by Proposition 3.8.5, we can find a continuous function g : Γ → C with eg = f.

(3.11.4)

By hypothesis, f admits local factorizations with respect to Γ. Hence f admits local factorizations with respect to Γ. By Lemma 3.11.3, this implies that g locally splits additively with respect to Γ. Therefore, from Theorem 3.7.3 we get a continuous function g+ : D+ → C, which is holomorphic in D+ , and a continuous function g− : D− ∪ {∞} → C, which is holomorphic in D− ∪ {∞}, such that g = g+ + g− on Γ. Then on Γ. f = eg+ eg− Hence, by definition of f, f=

z κ eg+ eg− . (z − p1 )κ1 . . . (z − pm )κm

Setting f− = eg−

and f+ =

(z − p1 κ

)κ1

eg+ . . . (z − pm )κm

we get the required factorization f = z f+ f− of f with respect to Γ.

3.12

Comments

In writing this chapter we used different sources. For sections 3.1–3.5, 3.8 and 3.9 see [Mu], [GKru], for Section 3.6 see [Bar]. The material of sections 3.7 and 3.11, in this form, probably appears here for the first time. For Section 3.10 see [K].

Chapter 4

The Rouché theorem for operator functions In this chapter we generalize to finite meromorphic Fredholm operator functions the classical Rouché theorem from Complex analysis and the Smith factorization form. The proof is based on the local Smith form for matrix functions.

4.1

Finite meromorphic Fredholm functions

In this section E is a Banach space. 4.1.1 Definition. Let w ∈ C, let U be a neighborhood of w, and let A : U \ {w} → L(E) be a holomorphic function which is meromorphic on U (Section 1.10.6). Then we shall say that A is finite meromorphic at w if the Laurent expansion of A at w is of the form A(z) =

∞

(z − w)n An ,

n=m

where (if m < 0) the operators Am , . . . , A−1 are finite dimensional. If, in addition, A0 is a Fredholm operator, then A is called finite meromorphic and Fredholm at w. The index of A0 then will be called the index of A at w. 4.1.2 Theorem. Let w ∈ C, and let W be a neighborhood of w. Let A : W \ {w} → L(E) be a holomorphic function which is finite meromorphic and Fredholm at w. Assume the index of A at w is zero. Then there exist a neighborhood U ⊆ W of w and a finite dimensional projector1 P in E such that, with Q := I − P , the following holds: 1 By

a projector in E we always mean a bounded linear projector in E, i.e., an operator P ∈ L(E) with P 2 = P . A projector P in E is called finite dimensional if dim Im P < ∞.

98

Chapter 4. The Rouché theorem for operator functions

There exist holomorphic functions S, T : U → GL(E) and a holomorphic function AP : U \ {w} → L(Im P ), which is meromorphic at w, such that on U \ {w}.

SAT = Q + P AP P Proof. Let A(z) =

∞

(4.1.1)

(z − w)n An

n=m

be the Laurent expansion of A at w. Since A0 is a Fredholm operator with index zero, by multiplication by an invertible operator (from the left or from the right), we may assume that A0 is a projector with dim Ker A0 < ∞. Using again that the operators Am , . . . , A−1 are finite dimensional, we can find finite dimensional projectors P and P0 in E such that P0 P = P P0 = P0 ,

(4.1.2)

A0 = Q + P0 ,

(4.1.3)

and, with Q := I − P , Aj Q = QAj = 0

for m ≤ j ≤ −1 .

(4.1.4)

Choose a neighborhood U ⊆ W of w so small that the Laurent expansion of A at w converges on U \ {w}. Setting A+ (z) = Q + P0 +

∞

(z − w)n An

and V (z) = A+ (z)Q + P

n=1

for z ∈ U , we get holomorphic functions A+ , V : U → L(E). Then, by (4.1.3) and (4.1.4), A+ Q = AQ on U \ {w} (4.1.5) and hence V Q = AQ on U \ {w}.

(4.1.6)

Since A+ (w) = Q + P0 , it follows from (4.1.3) and (4.1.2) that V (w) = I. By shrinking U we can achieve that V (z) ∈ GL(E)

for all z ∈ U .

(4.1.7)

Then it follows from the definition of V that V −1 P = P

on U,

(4.1.8)

and from (4.1.6) it follows that Q = V −1 AQ on U \ {w}.

(4.1.9)

4.1. Finite meromorphic Fredholm functions Setting

99

S = I − QV −1 A+ P

and

AP = P V −1 AP

on U

on U \ {w},

we obtain a holomorphic function S : U → L(E) and AP : U \ {w} → L(Im P ). The values of S are invertible, namely: S −1 = I + QV −1 A+ P

on U.

Moreover, V −1 AS = V −1 AQ + V −1 AP − V −1 AQV −1 A+ P

on U \ {w}.

By (4.1.9) this implies V −1 AS = Q + V −1 AP − QV −1 A+ P = Q + P V −1 AP + QV −1 AP − QV −1 A+ P = Q + P AP P + QV

−1

(A − A+ )P

(4.1.10)

on U \ {w}.

It follows from (4.1.4) that −1

A(z) − A+ (z) =

(z − w)An ,

z ∈ U \ {w} .

n=m

Since, by (4.1.4), An P = P An for m ≤ n ≤ −1, this implies that (A − A+ )P = P (A − A+ )P and therefore QV −1 (A − A+ )P = QV −1 P (A − A+ )P

on U \ {w}.

Since, by (4.1.8), V −1 P = P , this further implies that QV −1 (A − A+ )P = 0. Together with (4.1.10) this gives V −1 AS = Q + P AP P

on U \ {w}.

(4.1.11)

Setting T = V −1 , we get the required relation (4.1.1). It remains to observe that from (4.1.1) it follows that AP is meromorphic at w and holomorphic on U \ {w}, because A has these properties. The function AP in (4.1.1) can be represented by a meromorphic matrix function. Since the inverse of such a function is again meromorphic (if it exists), we immediately obtain the following

100


4.1.3 Corollary (to Theorem 4.1.2). Let D ⊆ C be an open set, and let Z ⊆ D be a discrete and closed subset of D. Let A be a holomorphic GL(E)-valued function on D \ Z which is finite meromorphic and Fredholm at each point of Z. Then A−1 is finite meromorphic and Fredholm at each point of Z. 4.1.4 Proposition. Let D ⊆ C be a connected open set, and let Z ⊆ D be a discrete and closed subset of D. Let A be a holomorphic L(E)-valued function on D \ Z which is finite meromorphic and Fredholm at each point of D. Suppose, there exists at least one point z0 ∈ D \ Z such that A(z0 ) is invertible. Then there exists a discrete and closed subset Z of D with Z ⊇ Z such that A(z) is invertible for each z ∈ D \ Z , and the function A−1 : D \ Z → GL(E) is finite meromorphic and Fredholm at each point of D. Proof. Denote by D the set of all w ∈ D \Z such that there exists a neighborhood U ⊆ D with U \ {w} ⊆ D \ Z and A(z) ∈ GL(E) for all z ∈ U \ {w}. We claim that D = D. Obviously, D is open and, by hypothethis, D ⊇ {z0 } = ∅. Since D is connected, it remains to prove that D is relatively closed in D. Let w ∈ D be a boundary point of D . Since, by hypothesis, A is finite meromorphic and Fredholm at w, first we can find a neighborhood W ⊆ D\Z such that A is finite meromorphic and Fredholm at w. Then from Theorem 4.1.2 we get a neighborhood U ⊆ W of w, a finite dimensional projector P in E, holomorphic functions S, T : U → GL(E) and a holomorphic function AP : U \{w} → L(Im P ), which is meromorphic at w, such that, with Q := I − P , A = S(Q + P AP P )T

on U \ {w}.

(4.1.12)

Then the determinant det AP is holomorphic on U \ {w} and meromorphic at w. Hence, there is a neighborhood V ⊆ U of w such that either det AP ≡ 0

on V \ {w}

(4.1.13)

or det AP (z) = 0

for all z ∈ V \ {w}.

(4.1.14)

By (4.1.12), from (4.1.13) it would follow that A is not invertible for all z ∈ V \{w}. But this is impossible, as w is a boundary point of D . Hence, we have (4.1.14). Again by (4.1.12) this means that A(z) is invertible for all z ∈ V \{w}, i.e., w ∈ D . Let Z be the set of all w ∈ D such that either w ∈ Z or w ∈ D \ Z and A(w) is not invertible. Since D = D , it follows, by definition of D , that Z is discrete and closed in D. 4.1.5 Proposition. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary. Let Z ⊆ D be a finite set, and let M : D \ Z → L(E) be a holomorphic function which is finite meromorphic at each point of Z, such that

M (z) < 1

for all z ∈ ∂D.

(4.1.15)

4.2. Invertible finite meromorphic Fredholm functions

101

Then I + M is finite meromorphic and Fredholm at each point of D, and the index of I + M is zero at each point of D. Moreover, there exists a finite subset Z of D with Z ⊇ Z such that A(z) is invertible for each z ∈ D \ Z , Proof. Since M is finite meromorphic at each point of Z, it is clear that I + M is finite moromorphic at each point of Z. Let K the sum of the principal parts of M at the points of Z, and set A = I + M − K on D \ Z. Then A admits a holomorphic extension to D, which we also denote by A. Let F(E) be the ideal of finite dimensional operators in E, and let F ∞ (E) be the closure of F(E) in L(E) with respect to the operator norm. Consider the ! factor algebra L(E) := L(E)/F ∞ (E). For T ∈ L(E), we denote by T! the class of ! ! ≡ 0, and it follows from (4.1.15) that T in L(E). Then K ! − I ! = M $(z) ≤ M (z) < 1

A(z)

for all z ∈ ∂D.

Hence, by the maximum principle, ! − I ! κ p ,

(4.3.1)

: U → GL(n, C), F, F : U → and such that, for some holomorphic functions E, E GL(m, C) in a neighborhood U ⊆ D of z0 , EAF =

Δ 0 0 0

F = and EA

0 Δ 0 0

(4.3.2)

are the r × r diagonal matrices with the diagonals where Δ and Δ (z − z0 )κ1 , . . . , (z − z0 )κr

and (z − z0 )κ 1 , . . . , (z − z0 )κ r ,

respectively. Since κ1 ≥ . . . ≥ κp and κ 1 ≥ . . . ≥ κ p , then it follows from (4.3.1) that κν > κ μ for 1 ≤ ν ≤ p ≤ μ ≤ r. (4.3.3) 2 This vector of powers is not identical with the numerical characteristic of A at w which we introduce below (Def. 11.3.6), but it is closely related: The vector of powers and the numerical characteristic uniquely determine each other (see the beginning of Section 11.5 for this relation.)

4.3. Smith factorization

107

Moreover it follows from (4.3.2) that −1 Δ 0 −1 −1 Δ 0 F−1 F =A=E E 0 0 0 0 and therefore

−1 Δ 0 = Δ 0 F−1 F EE 0 0 0 0

on U \ {z0 } .

(4.3.4)

−1 , and βμν the elements of F−1 F , where μ is the Let αμν be the elements of EE row index and ν is the column index. Then by (4.3.2) (z − z0 )κν αμν = (z − z0 )κ μ βμν

on U \ {z0 }

if 1 ≤ μ, ν ≤ r .

By (4.3.3) this implies that βμν (z0 ) = 0 for ν ≤ p ≤ μ ≤ r. This is impossible, because F−1 (z0 )F (z0 ) is invertible. Existence: Let M be the ring of germs of scalar meromorphic functions at z0 , and let O be the subring of holomorphic germs. Let M(k) be the algebra of k × kmatrices with elements from M, and let O(k) be the subalgebra matrices with elements from O. Denote by GO(k) the group of invertible elements in O(k). Let [z − z0 ] be the element in O defined by the function z − z0 . Further, let M(n, m) be the space of n × m-matrices with elements from M. Then A can be viewed as an element of M(n, m) and we have to find E ∈ GO(n) and F ∈ GO(m) such that, for some integers κ1 ≥ . . . ≥ κr , EAF is of the block form such that EAF is of the block form Δ 0 EAF = 0 0 where Δ is the r × r diagonal matrix with the diagonal [z − z0 ]κ1 , . . . , [z − z0 ]κr . Consider the following three row operations, which we will apply to matrices from M(n, m): (I) Multiplying a row by an element g ∈ GO. (II) Interchanging two rows. (III) Taking two different rows a and b and an arbitrary element f ∈ O and replacing the row b by b + f a. Operation (I) can be realized multiplying from the left by the diagonal matrix obtained from the unit matrix after replacing the corresponding element of the diagonal by g. Operation (II) can be realized multiplying from the left by the matrix obtained from the unit matrix after interchanging the corresponding columns.

108


Operation (III) can be realized multiplying from the left by the matrix obtained from the unit matrix after replacing one of the zero elements outside the diagonal by f . All these matrices belong to the group GO(n). Multiplying the corresponding matrices of the group GO(m) from the right we obtain the corresponding column operations, which we denote by (I ), (II ) and (III ), respectively. Therefore it is sufficient to prove that, by a finite number of the operations (I), (II), (III), (I ), (II ) and (III ), the matrix A can be transformed to a matrix of the required diagonal form. If a ∈ M and a = 0, then there is a uniquely determined entire number N such that the Laurent series of a is of the form a=

∞

an [z − z0 ]n

with

aN = 0.

n=N

We call this number the order of a and denote it by ord a. For the zero element 0 ∈ M we write ord 0 = ∞. For a ∈ M with a = 0, we define ! a = [z − z0 ]− ord a a. Then ! a ∈ GO for all a ∈ M with a = 0. Denote by N1 the minimal order of the elements of A. Since A is not the zero matrix, N1 < ∞. Choose 1 ≤ p ≤ n and 1 ≤ q ≤ m such that at the place (p, q) of A we have an element of order N1 . Denote this element by a. Now we proceed as follows: 1. We multiply the p-th row of A by ! a−1 . 2. We interchange the q − th column of the obtained matrix with the first column. 3. We interchange the p − th row of the now obtained matrix with the first row. After these operations, the orders of all elements of the matrix are still ≥ N1 , and at the place (1, 1) we have the element [z − z0 ]N1 . Now consider the element b at the place (1, 2) (first row and second column). Since ord b ≥ N1 , then [z − z0 ]ord b−N1 !b belongs to O and we may multiply the first column by [z − z0 ]ord b−N1 !b and then substract the result from the second column. So we get a matrix with a zero at the place (1, 2). Doing the same with the elements at the places (1, 3), . . . , (1, m), we end up with a matrix with [z − z0 ]N1 at the place (1, 1) and with zeros at the places (1, 2), . . . , (1, m). Then the same procedure with row operations leads to a matrix of the form ⎞ ⎛ [z − z0 ]N1 0 ... 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ . B 0 where B is an (n − 1) × (m − 1)-matrix with elements from M and the orders of all elements of B are still ≥ N1 . If B = 0, the proof is complete. If not we apply

4.3. Smith factorization

109

the same procedure to the matrix B. If N2 is the of B, then this gives a matrix of the form ⎛ [z − z0 ]N1 0 0 N2 ⎜ 0 [z − z ] 0 0 ⎜ ⎜ 0 0 ⎜ ⎜ .. .. ⎝ . . 0

minimal order of the elements ... ... C

⎞ 0 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0

where C is an (n − 2) × (m − 2)-matrix. If C = 0, we set κ1 = N2 and κ2 = N1 , interchange the first two columns and then the first two rows, and the proof is complete. If not we proceed in this way and, for some 3 ≤ r ≤ min(n, m), we end up with a block matrix of the form Δ 0 0 0 where Δ is the r × r diagonal matrix with the diagonal (z − z0 )N1 , . . . , (z − z0 )Nr .

This completes the proof of Lemma 4.3.1.

4.3.3 Corollary. Let w ∈ C, and let K : C \ L(n, C) be a rational matrix function of the form −1 (z − w)n Kn , (4.3.5) K(z) = n=−m

where K−m , . . . , K−1 , 1 ≤ m < ∞, are constant complex n × n-matrices. Then there exists a neighborhood U of w and a holomorphic matrix function A : U → L(n, C) such that A(z) is invertible for all z ∈ U \ {w} and K is the principal part of the Laurent expansion of A−1 at w. Proof. For K ≡ 0 this is trivial. Let K ≡ 0. Then, by the Smith factorization Theorem 4.3.1, there are integers κ1 ≥ . . . ≥ κr , 1 ≤ r ≤ n, a neighborhood U of w and holomorphic functions E, F : U → GL(n, C), such that K = EΔF where Δ is the n × n diagonal matrix with the diagonal (z − w)κ1 , . . . , (z − w)κr , 0, . . . , 0 . + ,- . (n−r) times

Let Δ+ be the n × n-diagonal matrix with the diagonal d1 , . . . , dn where if 1 ≤ j ≤ r and κj < 0 , (z − w)−κj dj = 1 if r + 1 ≤ j ≤ n or 1 ≤ j ≤ r and κj ≥ 0 .

110


Then Δ+ is holomorphic on U , invertible on U \ {w}, and Δ−1 + − Δ extends holomorphically to w. Set A = F −1 Δ+ E −1 . Then A is holomorphic on U , invertible on U \ {w}, and A−1 (z) = E(z)Δ−1 + (z)F (z)

(z) − Δ(z) F (z) = E(z)Δ(z)F (z) + E(z) Δ−1 +

= K(z) + E(z) Δ−1 + (z) − Δ(z) F (z) for z ∈ U \ {w}.

Since E(Δ−1 + − Δ)F extends holomorphically to w, this shows that K is the principal part of the Laurent expansion of A−1 at w. It is impossible to replace in Corollary 4.3.3 the prescribed function (4.3.5) by an arbitrary function of the form 0

K(z) =

(z − w)n Kn ,

n=−m

with matrices K−m , . . . , K0 , 1 ≤ m < ∞ where K0 = 0. We give a counterexample: 4.3.4 Example. Let

K(z) :=

1 0

z −1 . 1

(4.3.6)

Then it is impossible to find a neighborhood U of 0 ∈ C and holomorphic functions A, B : U → L(2, C) such that A(z) is invertible for z ∈ U \ {0} and 1 z −1 −1 A (z) = + zB(z) , z ∈ U \ {0}. 0 1 Indeed, assume this is possible. Then 1 0 1 z −1 = A(z) + zB(z)A(z) , 0 1 0 1

Let A(z) =

a11 (z) a12 (z) a21 (z) a22 (z)

z ∈ U \ {0}.

and B(z) =

(4.3.7)

b11 (z) b12 (z) . b21 (z) b22 (z)

Then it follows from (4.3.7) that 0 = a12 (z) + z −1 a22 (z) + zb11 (z)a12 (z) + zb12 (z)a22 (z) and 1 = a22 (z) + zb21 (z)a12 (z) + zb22 (z)a22 (z) for z ∈ U \ {0}. This is impossible, as from the first relation it follows that a22 (0) = 0 and from the second relation it follows that a22 (0) = 1.

4.4. The Rouché theorem

4.4

111

The Rouché theorem

4.4.1 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, let Z ⊆ D be a finite set, and let A : D \ Z → GL(E) be a holomorphic function which is finite meromorphic and Fredholm at the points of Z. Then ind∂D A (Def. 4.2.4) is an integer. Proof. By Cauchy’s theorem, we may assume that Z consists only of a single point w. Since, on D \ {w}, the values of A are invertible and A is finite meromorphic and Fredholm at w, it is clear that the index of A at w is zero. Therefore, by Theorem 4.1.2, there exist a neighborhood U ⊆ D of w, a finite dimensional projector P in E, holomorphic functions S, T : U → GL(E) and a holomorphic function AP : U \ {w} → L(Im P ), which is meromorphic at w, such that, with Q := I − P , A = S(Q + P AP P )T on U \ {w}. (4.4.1) We may assume that U is a disc. Then, again by Cauchy’s theorem, ind∂D A = ind∂U A . Since ind∂U T = ind∂U S = 0, this further implies, by (4.4.1) and Proposition 4.2.5, that ind∂D A = ind∂U (Q + P AP P ). By Cauchy’s theorem this implies that ind∂D A = ind∂U AP .

(4.4.2)

By the Smith factorization Theorem 4.3.1, there exist uniquely determined integers κ1 ≥ . . . ≥ κr , a neighborhood U ⊆ W of w and holomorphic functions E, F : U → GL(Im P ) such that Δ 0 (4.4.3) EAP F = 0 0 where, with respect to an appropriate basis of Im P , Δ can be represented by the r × r diagonal matrix with the diagonal (z − w)κ1 , . . . , (z − w)κr . Note that 1 ind∂U Δ = 2πi

tr(Δ Δ

−1

1 ) dz = 2πi

∂U

r ∂U j=1

κj κj . dz = z−w j=1 r

By Proposition 4.2.5 and (4.4.3), this implies that ind∂U AP =

r

κj .

j=1

Together with (4.4.2) this proves the theorem

112


4.4.2 Lemma. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary. Let Z ⊆ D be a finite set, and let M : D \ Z → L(E) be a holomorphic function which is finite meromorphic at each point of Z, such that

M (z) < 1

for all z ∈ ∂D.

(4.4.4)

Then I + M is finite meromorphic and Fredholm at each point of D, and the index of I + M is zero at each point of D. Moreover, there exists a finite subset Z of D with Z ⊇ Z such that I + M (z) is invertible for each z ∈ D \ Z . Moreover, ind∂D (I + M ) = 0 .

(4.4.5)

Proof. Except for relation (4.4.5), this is precisely the statement of propositon 4.1.5. Moreover, by the same Proposition 4.1.5, for each 0 ≤ t ≤ 1, the function I + tM has the same properties. Obviously, the function [0, 1] t −→ ind∂D (I + tM ) is continuous. Since, by Theorem 4.4.1, the values of this function are integers and since ind∂D I = 0, it follows that ind∂D (I + M ) = 0. 4.4.3 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, let Z ⊆ D be a finite set, let A : D \ Z → GL(E) be a holomorphic function which is finite meromorphic and Fredholm at the points of Z, and let S : D \ Z → GL(E) be a holomorphic function which is finite meromorphic at each points of Z such that

A−1 (z)S(z) < 1 for z ∈ ∂D . (4.4.6) Then the function A + S is finite meromorphic and Fredholm at each point of Z, and ind∂D (A + S) = ind∂D A . (4.4.7) Proof. By Lemma 4.4.2, the function I +A−1 S is finite meromorphic and Fredholm at each point of D, the index of I + A−1 is zero at each point of D, there exists a finite subset Z of D with Z ⊇ Z such that I + A−1 S(z) is invertible for each z ∈ D \ Z , and ind∂D (I + A−1 S) = 0 . Since A + S = A(I + A−1 S), by Proposition 4.2.5, this implies ind∂D (A + S) = ind∂D A + ind∂D (I + A−1 S) = ind∂D A .

4.5

Comments

The material of this chapter is mostly taken from [GS].

Chapter 5

Multiplicative cocycles (OG-cocycles) Let A be a Banach algebra with unit 1, and let G be an open subgroup of the group of invertible elements of A. The Runge approximation Theorem 2.5.6 for invertible scalar functions admits the following generalization: 5.0.1 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary (possibly not connected), and let f : D → G be a continuous function which is holomorphic in D, such that all values of f belong to the same connected component of G. (i) If C\D is connected, then f can be approximated uniformly on D by G-valued functions defined and holomorphic on C. (ii) Suppose C \ D is not connected. Let U1 , . . . , UN be the bounded connected components of C \ D, and assume that, for each 1 ≤ j ≤ N , a point pj ∈ Uj is given. Then f can be approximated uniformly on D by G-valued functions defined and holomorphic on C \ {p1 , . . . , pN }. Part (i) of this theorem will be proved at the end of Section 5.4, and part (ii) will be obtained in Section 5.10 as a special case of Theorem 5.10.5. (Theorem 5.10.5 is the formulation of Theorem 5.0.1 on the Riemann sphere.) Theorem 5.0.1 is one of the main statements of the theory developed in the present chapter. The proof is much more difficult than the proof of Theorem 2.5.6, and can be obtained only in the framework of the theory of OG -cocycles, developed in this chapter.1 The reason is that the group G, in general, is not commutative. After some preparation in Section 5.3, in Section 5.4 we only prove part (i) of Theorem 5.0.1. Then, in Section 5.5, we prove the Cartan lemma, which can be viewed as a special case of Theorem 0.0.1 stated in the introduction to this book. 1 At

least the authors do not see another way.

Chapter 5. Multiplicative cocycles (OG -cocycles)

114

Theorem 0.0.1 in turn can be considered as a special result on the triviality of OGL(E) -cocycles (see Definition 5.6.1 below), because a GL(E)-valued function, defined and holomorphic in the intersection of two open sets, may be viewed as a special OGL(E) -cocycle. Using the language introduced in Definition 5.6.1, Theorem 0.0.1 then says that this cocycle is OGL(E) -trivial. To prove Theorem 0.0.1, however we have to pass to general multiplicative cocycles (Def. 5.6.1). In Section 5.6 we introduce the language of multiplicative cocycles and state the main result on such cocycles. This is Theorem 5.6.3, which contains Theorem 0.0.1 as a special case. In Section 5.9 we prove it in the simply connected case, whence also Theorem 0.0.1 is proved in the case when D1 ∪ D2 is simply connected. Using this, then in Section 5.10 we prove the Runge approximation Theorem 5.0.1 in its general form. At the end, using the Runge approximation Theorem 5.0.1 in its general form, in Section 5.11 we prove Theorem 5.6.3 in its general form, whence also Theorem 0.0.1 is proved in its general form. In Section 5.13 we prove the generalized Weierstrass Theorem 0.0.2, stated in the introduction to this book, as well as the corresponding right-sided and twosided versions. In Section 5.12 we prove Weierstrass theorems for functions of the form I + K, where the values of K are compact.

5.1

Topological properties of GL(E)

5.1.1. Until now it was not important whether the group of invertible elements of a Banach algebra is connected or not. This changes beginning with the present chapter. Therefore we devote this section to this question. First recall the following example of a Banach algebra with a non-connected group of invertible elements: Let E be an infinite dimensional Banach space, and let F ω (E) be the ideal of compact operators in L(E). Then the group of invertible elements of L(E)/F ω (E) is not connected, because this group is the image under the canonical map L(E) −→ L(E)/F ω (E) of the set of Fredholm operators in L(E) (see, e.g., [GGK2]), where Fredholm operators with different indicies define different connected components of the group G L(E)/F ω (E) . For many Banach spaces, the group of invertible operators is connected. This is the case, for example, for all Hilbert spaces, which we prove below (Theorem 5.1.5). Note without proofs that for each of the following Banach spaces the group of invertible operators is not only connected but even contractible: Hilbert spaces of infinite dimension, c0 , lp , 1 ≤ p ≤ ∞,

5.1. Topological properties of GL(E)

115

Lp [0, 1], 1 ≤ p ≤ ∞, C 0 (K) (K a compact metric space of continuum cardinality), C k (M ), k ∈ N, (M a smooth compact manifold). The proofs and more examples can be found in the original papers [A, Ku, N1, Mi, MiE, EMS1, EMS2]. The interested reader is recommended to consult also the review about these papers written by G. Neubauer [N2] . But there are also quite natural Banach spaces with non-connected groups of invertible operators. At the end of this section, we give an example. We begin with the following simple lemma. 5.1.2 Lemma. Let E be a Banach space, let A ∈ GL(E), and let σ(A) be the spectrum of A. Assume there exists a continuous curve γ : [0, 1] → C \ σ(A) such that γ(0) = 0 and |γ(1)| > A . Then A belongs to the connected component of the unit operator in GL(E). Proof. Since γ(t) ∈ σ(A) and therefore A − γ(t) ∈ GL(E) for all 0 ≤ t ≤ 1, the operators A = A − γ(0)I and A − γ(1)I belong to the same connected component of GL(E). Since |γ(1)| > A , setting t , 0 ≤ t ≤ 1, α(t) = −γ(1) I − γ(1) we obtain a continuous curve α : [0, 1] → GL(E) which connects the operator A − γ(1)I = α(1) with −γ(1)I = α(0). As −γ(1)I and I belong to the same connected component of GL(E), this completes the proof. 5.1.3 Corollary. For each n ∈ N∗ , the group GL(n, C) of invertible complex n × nmatrices is connected. Proof. This follows from Lemma 5.1.2, because the spectrum of a matrix is finite. 5.1.4. Now let H be a Hilbert space of infinite dimension. As already mentioned above, then GL(H) is even contractible [Ku] (in distinction to the groups GL(n, C), n ∈ N∗ ). In the present book we need only the simpler fact that GL(H) is connected.2 Again using Lemma 5.1.2, this can be easily deduced from the spectral representation of unitary operators: 5.1.5 Theorem. For each Hilbert space H, the group GL(H) of invertible operators is connected. 2 This is due to the fact that we deal with functions of one complex variable and that the topology of a domain in the complex plane is simple compared to the topology of domains in higher dimensional spaces. That GL(H) is even contractible becomes important if we pass to several complex variables (see [Bu]).

116


Proof. Let GLI (H) be the connected component of the unit operator in GL(H). We have to prove that GL(H) = GLI (H). Let A ∈ GL(H) be given. Then A = U S, where S := (A∗ A)1/2 is positive definite, and U := AS −1 is unitary. Since the spectrum σ(S) of S is contained in the real line and 0 ∈ σ(S), it follows from Lemma 5.1.2 that S ∈ GLI (H). It remains to prove that U ∈ GLI (H). It follows from the spectral representation of unitary operators (see, e.g., [GGK1]) that H can be written as an orthogonal sum H = H1 ⊕ H2 , where H1 and H2 are invariant subspaces of U such that the spectrum of U H lies on the half circle |z| = 1, Im z ≥ 0, and 1 the spectrum of U H lies on the half circle |z| = 1, Im z < 0. Therefore, by 2 Lemma 5.1.2, U H belongs to the connected component of the unit operator of 1 H1 , and U H2 belongs to the connected component of the unit operator of H2 . Hence U ∈ GLI (H). Next we give an example for a Banach space E such that GL(E) is not connected: 5.1.6 Theorem. 3 Let c0 be the Banach space of sequences ξ = {ξn }n∈N of complex numbers tending to zero with the norm

ξ ∞ = max |ξn |, n∈N

and let lp , 1 ≤ p < ∞, be the Banach space of sequences {ξn }n∈N of complex numbers with the norm 1/p ∞

ξ p = |ξn |p < ∞. n=0

Then the group GL(c0 ⊕lp ) consists of an infinite number of connected components. To prove this theorem, we need the following three lemmas: 5.1.7 Lemma. Let c0 and lp , 1 ≤ p < ∞ be as in Theorem 5.1.6, and let Pn be the projector in c0 , defined by

(5.1.1) Pn {ξn }n∈N = (ξ1 , . . . , ξn , 0, . . .). For j ∈ N, we denote by fj : lp → C the functional defined by fj {ξn }n∈N = ξj . Let A ∈ L(c0 , lp ) and j ∈ N be fixed. Then lim fj A(I − Pn ) = 0 n→∞

(5.1.2)

3 This example is due to A. Douady [Do1]. As observed in [Do1] instead of c and l here one p 0 could take an arbitrary pair of Banach spaces F and G such that each operator from L(F, G) is compact, and F and G are isomorphic to their hyperplanes.

5.1. Topological properties of GL(E) where

fj A(I − Pn ) :=

117

sup ξ∈c0 ,ξ∞ ≤1

fj A(ξ − Pn ξ) .

Proof. Assume the contrary. Since fj A(I − Pn ) ≥ fj A(I − Pn+1 ) then there exists ε > 0 such that fj A(I − Pn ) > ε

for all n ∈ N,

for all n ∈ N.

(5.1.3)

Choose a number n1 ∈ N (arbitrarily). Then, by (5.1.3), we can find a vector η(1) ∈ Im(I − Pn1 ) with η (1) ∞ ≤ 1 and (5.1.4) η (1) ) > ε. (fj A)( Since η(1) tends to zero, we have lim (I − Pn ) η (1) ∞ = 0. n→∞

Therefore it follows from (5.1.4) that there exists n2 ∈ N so large that (fj A)(Pn2 η(1) ) > ε. Set η (1) =

(fj A)(Pn2 η(1) ) (fj A)(Pn2 η(1) )

Pn2 η(1) .

Then η (1) ∈ Im(Pn2 − Pn1 ),

η (1) ∞ ≤ 1

and

(fj A)(η (1) ) > ε.

If this way we get by induction an increasing sequence nμ ∈ N, μ ∈ N∗ , and a sequence of vectors η (μ) ∈ c0 , μ ∈ N∗ , such that η (μ) ∈ Im(Pnμ+1 − Pnμ ),

η (μ) ∞ ≤ 1

and

(fj A)(η (μ) ) > ε

for all μ. Now, for each N ∈ N∗ , we set φ(N ) = η (1) + . . . + η (N ) . Then

φ(N ) ∞ = max η (μ) ∞ ≤ 1 1≤μ≤N

and (fj A)(φ(N ) ) = (fj A)(η (1) ) + . . . + (fj A)(η (N ) ) > N ε. This is a contradiction to the boundedness of fj A.


118

5.1.8 Lemma. Let c0 and lp , 1 ≤ p < ∞, be as in Theorem 5.1.6. Then each operator A ∈ L(c0 , lp ) can be approximated in the operator norm by finite dimensional operators, namely if Pn is the projector in c0 defined by (5.1.1), then lim A − APn = 0.

(5.1.5)

n→∞

Proof. Assume that there exists a bounded linear operator A : c0 → lp such that (5.1.5) is violated. Since

A − APn ≥ A − APn+1

for all n ∈ N,

then there exists ε > 0 with

A − APn > ε

for all n ∈ N.

(5.1.6)

Let A be the operator norm on A. Choose α ∈ c0 with

α ∞ ≤ 1

and

A α p > A −

ε . 10

(5.1.7)

α − Pn α = 0. Therefore and by (5.1.7) we can As α tends to zero, then limn→∞ find m ∈ N with ε p > A − .

APm α 10 . Then Set α = Pm α α ∈ Im Pm ,

α ∞ ≤ 1

and

Aα p > A −

ε . 10

(5.1.8)

Let Rn : lp → lp be the projector defined by Rn {ξn }n∈N = ξj . Since 1 ≤ p < ∞, then limn→∞ (I − Rk )Aα p = 0. Hence, we can find k ∈ N∗ such that ε . (5.1.9)

(I − Rk )Aα p ≤ 10 Together with (5.1.8) this implies that ε

Rk Aα p ≥ A − . 5

(5.1.10)

From Lemma 5.1.7 it follows that limn→∞ Rk A(I − Pn ) = 0. Therefore, we can find an integer l > m with

Rk A(I − Pl ) ≤

ε . 10

(5.1.11)

5.1. Topological properties of GL(E)

119

By (5.1.6) there exists β ∈ Im(I − Pl ) such that

β ∞ ≤ 1

and

Aβ p ≥ ε.

(5.1.12)

ε 10

(5.1.13)

Together with (5.1.11) this implies that

Rk Aβ p ≤ and

ε . 10 Since α ∈ Im Pm , β ∈ Im(I − Pl ) and l > m, now we have

α + β ∞ = max α ∞ , β ∞ ≤ 1,

(I − Rk )Aβ p ≥ ε −

(5.1.14)

and from (5.1.9), (5.1.10), (5.1.13) and (5.1.14) it follows that ε

A(α +β) p ≥ Rk Aα p − (I −Rk )Aα p + (I −Rk )Aβ p − Rk Aβ p ≥ A + . 2 This is a contradiction to the definition of A .

5.1.9 Lemma. Let c0 and lp , 1 ≤ p < ∞, be as in Theorem 5.1.6, and let P be the projector in c0 ⊕ lp with Im P = c0 and Ker P = lp . Set Q = I − P . Let A ∈ GL(c0 ⊕lp ). Then P Ac is a Fredholm operator in c0 and QAl is a Fredholm p 0 operator in lp . Proof. We have P = P AA−1 P = P AP A−1 P + P AQA−1 P and

P = P A−1 AP = P A−1 P AP + P A−1 QAP.

Since, by Lemma 5.1.8, the operators P AQA−1P and P A−1 QAP are compact, −1 this implies that both P AP A c and P A−1 P Ac are Fredholm operators. Hence 0 0 (see., e.g., [GGK2]), P Ac is a Fredholm operator. In the same way we see that 0 QAl is a Fredholm operator. p

Proof of Theorem 5.1.6. We consider the space c0 ⊕ lp as the space of families ∞ ξn h=−∞ , ξm ∈ C, such that lim ξn = 0

n→−∞

and

∞

|ξn |p < ∞.

n=0

Then, for k ∈ Z, we have the shift operator Ak defined by

= ξn+k }j=−∞∞ . Ak ξn }j=−∞∞


120

Clearly, then Ak ∈ GL(c0 ⊕ lp ) for all k ∈ Z. Moreover, if P is the projector from Lemma 5.1.9, P Ak c is a Fredholm operator with index k in c0 . 0 Assume Ak and Am belong to the same connected component of GL(c0 ⊕ lp ). Then Ak−m = Ak A−1 m belongs to the connected component of GL(c0 ⊕ lp ) which contains the unit operator. Hence, there is a continuous curve A : [0, 1] → GL(c0 ⊕ lp ) with A(0) = I and A(1) = Ak−m . Then, by Lemma 5.1.9, {P A(t)c }0≤t≤t , 0 is a continuous family of Fredholm operators in c0 . Since A(0) = I has the index zero, then also A(1) = Ak−m ck has the index zero. Hence k = m. 0 Hence we proved that, for all k, m ∈ Z with k = m, the operators Ak and Am belong to different connected components of GL(c0 ⊕ lp ).

5.2

Two factorization lemmas

The following lemma will be used several times in this book: 5.2.1 Lemma. Let A be a Banach algebra with unit 1 and the norm · , and let A1 , A2 be two algebraic subalgebras of A with 1 ∈ A1 ∩ A2 , which are Banach algebras with respect to their own norms · 1 and · 2 , respectively. Assume that: (i) x j ≥ x for each x ∈ Aj , j = 1, 2. (ii) Each element x ∈ A can be written in the form x = x1 + x2 with xj ∈ Aj , j = 1, 2. Let C < ∞ be the smallest constant such that, for all x ∈ A, there exist xj ∈ Aj with x = x1 + x2 and j = 1, 2. (5.2.1)

xj j ≤ C x , (Such a constant then exists by the Banach open mapping theorem.) Then, for each a ∈ A with

a
0 such that Uε (1) := x ∈ A 1 − x < ε ⊆ G. By Proposition 5.4.1, each element in G1 A is a finite product of elements from Uε (1). Since G is a subgroup of GA, it follows that G1 A ⊆ G, and hence aG1 A ⊆ G for all a ∈ G. Since Ga = aG1 for all a ∈ GA, this implies (5.4.2). 5.4.3. If A is a Banach algebra with unit 1, and GA is the group of invertible elements of A, then we define exp a = ea = and log a = −

∞ (1 − a)n n n=1

∞ an n! n=0

for all a ∈ A

for all a ∈ A with 1 − a < 1.

(5.4.3)

(5.4.4)

5.4.4 Lemma. Let A be a Banach algebra with unit 1, let GA be the group of invertible elements of A, and let G1 A be the connected component of GA which contains the unit element. Then: −1 (i) ea ∈ G1 A for all a ∈ A, where ea = e−a . (ii) If a ∈ A with a − 1 < 1, then elog a = a.

5.4. Runge approximation. First steps

127

Proof. If A is commutative, this is well known. In the general case, we can pass to the smallest closed subalgebra of A which contains a and the unit element. This subalgebra is commutative. 5.4.5 Definition. Let D ⊆ C be a bounded open set, let A be a Banach algebra with unit 1, and let GA be the group of invertible elements of A. Then we denote by OA (D) the algebra of continuous A-valued functions on D which are holomorphic in D. By setting

f OA (D) := max f (z) A ,

f ∈ OA (D),

z∈D

we introduce a norm in OA (D), where · A is the norm of A. In this way, also OA (D) becomes a Banach algebra with unit. If it is clear what we mean we simply write · instead of · OA (D) . If G is an open subgroup of GA, then we denote by OG (D) the subset of O (D) which consists of the functions with values in G. Note that then OGA (D) is the group of invertible elements of OA (D), and, if G is an open subgroup of GA, then OG (D) is an open subgroup of OGA (D). A

5.4.6 Proposition. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, and let P ⊆ C \ D be a set which contains at least one point of each bounded connected component of C \ D (if there is any5 ). Let A be a Banach algebra with unit 1, and let G be an open subgroup of the group of invertible elements of A. G Then, for each f ∈ O (D), the following two conditions are equivalent: (i) f can be approximated uniformly on D by functions from OG (C \ P ). (ii) There exists f ∈ OG (C \ P ) such that f and f belong to the same connected G component of O (D). Proof. Since the connected components of OG (D) are open, it is clear that (i) ⇒ (ii). Suppose (ii) is satisfied. Since f and f belong to the same connected component of OG (D), the function f f−1 belongs to the connected component of the unit element in OG (D). Therefore, by Proposition 5.4.1, it can be written in the form f f−1 = (1 + g1 ) . . . (1 + gn ) (5.4.5) where gj ∈ OA (D) with gj < 1. Set hj = log(1 + gj ) =

∞ (−1)ν+1 ν=1

5 If

C \ D is connected, then P = ∅ is possible.

ν

gjν ,

1 ≤ j ≤ n.


128

Since, for any j, the series on the right-hand side converges uniformly on D, so we obtain functions hj ∈ OA (D). By Lemma 5.4.4 (ii), 1 + gj = ehj and therefore, by (5.4.5), f = eh1 . . . ehn f. Now, by the Runge approximation Theorem 2.2.2, for each j, we can find a sequence (hj,ν )ν∈N which converges to hj uniformly on D. Then gν := eh1,ν . . . ehn,ν f,

ν ∈ N,

is a sequence of functions from OA (C\P ) which converges to f uniformly on D. By Lemma 5.4.4 (i) the values of each ehj,ν belong to G1 A, the connected component of GA which contains the unit element. Hence the values of g f−1 = eh1 . . . ehn belong to G1 A. Since the values of f belong to G, this means that also the values of g belong to G. Using the Riemann mapping theorem, now we prove: 5.4.7 Lemma. Let D ⊆ C be a bounded, simply connected open set (Section 1.4.3) with piecewise C 1 -boundary. Let A be a Banach algebra with unit 1, and let G1 A be the connected component of the unit element in the group of invertible elements G1 A of A. Then the group O (D) is connected (with respect to uniform convergence on D). G1 A

(D) be given. We have to find a continuous curve in Proof. Let some f ∈ O G1 A O (D) which connects f and the constant function with value 1. Since G1 A is an open subset of A, we have ε > 0 such that the ball A Bε (f ) := g ∈ O (D) f − g OA (D) < ε G1 A

is contained in O (D). Moreover, by the Mergelyan approximation Theorem 2.2.1, there exists a holomorphic function f : U → A defined in a neighborhood U of D such that f|D ∈ Bε (f ). Therefore it is sufficient to find a continuous curve G1 A in O (D) which connects f|D with the constant function with value 1. Since D is bounded, simply connected and with piecewise C 1 -boundary, after shrinking U , we may assume that also U is bounded and simply connected. Therefore, by the Riemann mapping theorem, we can find a biholomorphic map Φ from U onto the unit disc. Set λ(t) (z) := f Φ−1 (1 − t)Φ(z) for t ∈ [0, 1], z ∈ D. This defines a continuous curve λ : [0, 1] → OG1 A (D) with λ(0) = f|D . The function λ(1) has the constant value f Φ−1 (0) ∈ G1 A.

5.5. The Cartan lemma

129

we can find a continuous curve γ : [1, 2] → G1 A with Since G1A is connected, γ(1) = f Φ−1 (0) and γ(2) = 1. Setting for z ∈ D, λ(t) (z) if t ∈ [0, 1] , γ(t) (z) = γ(t) (z) if t ∈ [1, 2] , we get a continuous curve with the required properties.

Now we can prove part (i) of Theorem 5.0.1. Proof of Theorem 5.0.1 (i). Let G1 A be the connected component of the group of invertible elements of A which contains the unit element. Since all values of f belong to the same connected component of G, after multiplying by a constant element of G we may assume that f belongs to the group OG1 A (D). By the preceding Lemma 5.4.7, this group is connected. In particular, condition (ii) in Proposition 5.4.6 is satisfied (with P = ∅). Therefore it follows from this proposition that f can be approximated uniformly on D by functions from OG1 A (C).

5.5

The Cartan lemma

5.5.1 Definition. A pair (D1 , D2 ) of bonded open sets D1 , D2 ⊆ C with piecewise C 1 -boundaries will be called a Cartan pair if: – The intersection D1 ∩ D2 is not empty and has piecewise C 1 -boundary, and C \ D1 ∩ D2 is connected. – The union D1 ∪ D2 has piecewise C 1 -boundary. – D1 \ D2 ∩ D2 \ D1 = ∅. For example, if D1 , D2 ⊆ C are two open rectangles such that also D1 ∪ D2 and D1 ∩ D2 are non-empty rectangles, then (D1 , D2 ) is a Cartan pair in the sense of this definition. If (D1 , D2 ) are a Cartan pair, then the closures of the connected components of D1 ∩ D2 are pairwise disjoint. Thisfollows from the fact that D1 ∩ D2 has piecewise C 1 -boundary. Moreover, since D1 \D2 ∩ D2 \D1 = ∅, then D1 ∩ D2 = D1 ∩ D2 . In this section we prove the following 5.5.2 Lemma (Cartan lemma). Let (D1 , D2 ) be a Cartan pair, let A be a Banach algebra with unit 1, and let G be an open subgroup of the group of invertible elements of A. Further, let f ∈ OG (D1 ∩ D2 ) (Def. 5.4.5) such that all values of f belong to the same connected component of G. Then there exist fj ∈ OG (Dj ), j = 1, 2, such that f = f1 f2 on D1 ∩ D2 . (5.5.1) Moreover, then, for each ε > 0, the following two assertions are true:


130

(i) There exist fj ∈ OG (Dj ), j = 1, 2, with (5.5.1) and max f1 (z) − 1 < ε.

(5.5.2)

z∈D 1

(ii) There exist fj ∈ OG (Dj ), j = 1, 2, with (5.5.1) and max f2 (z) − 1 < ε.

(5.5.3)

z∈D 2

We prove this lemma in two steps. 5.5.3 Lemma. Let (D1 , D2 ) be a Cartan pair, and let A be a Banach algebra with unit 1. Then, for each ε > 0, there exists δ > 0 such that, for each g ∈ O(D1 ∩ D2 , A) (for the notation cf. 5.4.5) with max

z∈D 1 ∩D 2

g(z) < δ,

there exist gj ∈ O(Dj , A) with max gj (z) < ε ,

j = 1, 2,

z∈D j

such that 1 + g = (1 + g1 )(1 + g2 )

on D1 ∩ D2 .

A

Proof. We consider O (D1 ∩ D2 ), OA (D1 ) and OA (D2 ) as Banach algebras enA dowed with the maximum norm. Then, by Corollary 5.3.5, each f ∈ O (D1 ∩ D2 ) A can be written in the form f = f1 + f2 with fj ∈ O (Dj ). Therefore the assertion follows from Lemma 5.2.1. Proof of Lemma 5.5.2. Let G1 A be the connected component of the unit element in the group of invertible elements A. Since all values of f belong to the same connected component of G, by multiplication by a constant element, we may achieve that f (z) ∈ G1 A for all z ∈ W . Since D1 ∩ D2 has piecewise C 1 boundary and C \ D1 ∩ D2 is connected, we can apply part (i) of the Runge approximation Theorem 5.0.1 (which was proved at the end of Section 2.2.2). So, for each δ > 0, we can find fδ ∈ OG1 A (C) with max

z∈D 1 ∩D 2

f (z)fδ−1 (z) − 1 < δ

and

max

z∈D 1 ∩D 2

fδ−1 (z)f (z) − 1 < δ. (1)

(2)

Therefore, by Lemma 5.5.3, for sufficiently small δ, we can find gj , gj such that (k)

gj OA (D ) < ε , j = 1, 2, k = 1, 2,

A

∈ O (Dj )

j

and

(1) (1) f fδ−1 = 1 + g1 1 + g2

(2) (2) and fδ−1 f = 1 + g1 1 + g2 (1)

(1)

on D1 ∩ D2 . To prove assertion (i) we set f1 = (1 + g1 ) and f2 = (1 + g2 )fδ . To (2) (2) prove assertion (ii) we set f1 = fδ (1 + g1 ) and f2 = 1 + g2 .

5.6. OG -cocycles

131

5.6 OG -cocycles. Definitions and statement of the main result Here we introduce the notion of multiplicative cocycles and state the main result on them, which will be proved in the subsequent sections. Throughout this section, A is a Banach algebra with unit 1, and G is an open subgroup of the group of invertible elements of A. If U ⊆ C is a non-empty open set, then we denote by OG (U ) the group of holomorphic G-valued functions defined on U (as everywhere in this book). For practical reasons, we set OG (∅) = {1} where 1 is the unit element of A. 5.6.1 Definition. Let D ⊆ C an open set, and let U = {Uj }j∈I be an open covering of D. (i) We denote by Z 1 (U, C G ) the set of all families f = {fjk }j,k∈I of functions fjk ∈ C G (Uj ∩ Uk ) with on Uj ∩ Uk ∩ Ul ,

fjk fkl = fjl

(5.6.1)

for all j, k, l ∈ I with Uj ∩ Uk ∩ Ul = ∅. The elements of Z 1 (U, C G ) will be called (U, C G )-cocycles. Condition (5.6.1) is called the (multiplicative) cocycle condition. Note that it in particular implies that fjj = 1

and

−1 fjk = fkj .

(ii) We denote by Z 1 (U, OG ) the set of all cocycles f ∈ Z 1 (U, C G ) such that fjk ∈ OG (Uj ∩ Uk ) for all j, k ∈ I. The elements of Z 1 (U, OG ) will be called (U, OG )-cocycles. (iii) Two cocycles f, g ∈ Z 1 (U, C G ) will be called C G -equivalent or continuously equivalent if there exists a family {fj }j∈I of functions fj ∈ C G (Uj ) such that gjk = hj fjk h−1 k

on Uj ∩ Uk

for all j, k ∈ I with Uj ∩ Uk = ∅. (iv) Two cocycles f, g ∈ Z 1 (U, OG ) will be called OG -equivalent or holomorphically equivalent if there exists a family {fj }j∈I of functions fj ∈ OG (Uj ) such that on Uj ∩ Uk gjk = hj fjk h−1 k for all j, k ∈ I with Uj ∩ Uk = ∅. (v) A cocycle f ∈ Z 1 (U, C G ) will be called C G -trivial or continuously trivial if there exists a family {fj }j∈I of functions fj ∈ C G (Uj ) such that fjk = fj fk−1

on Uj ∩ Uk

132

Chapter 5. Multiplicative cocycles (OG -cocycles) for all j, k ∈ I with Uj ∩ Uk = ∅. In other words, f is called C G -trivial if it is C G -equivalent to the cocycle e ∈ Z 1 (U, C G ) defined by ejk ≡ 1, j, k ∈ I. In this case we say also that f splits continuously.

(vi) A cocycle f ∈ Z 1 (U, OG ) will be called OG -trivial or holomorphically trivial if there exists a family {fj }j∈I of functions fj ∈ OG (Uj ) such that fjk = fj fk−1

on Uj ∩ Uk

for all j, k ∈ I with Uj ∩ Uk = ∅. In this case we say also that f splits holomorphically. Due to P. Cousin the elements of Z 1 (U, OG ) are also called multiplicative Cousin problems. By Theorem 2.3.1, the additive Cousin problems have always a solution. This is no more true for multiplicative Cousin problems. We give an example: 5.6.2 Example. Let 0 < r < R < ∞. Denote by D the annulus D := z ∈ C r < |z| < R , which we cover by the open sets r U1 := z ∈ D Im z < 2

and

r U2 := z ∈ D Im z > − . 2

Then the intersection U1 ∩ U2 consists of two connected components V1 := z ∈ U1 ∩ U2 Re z < 0 and

V2 := z ∈ U1 ∩ U2 Re z > 0 .

Take a Banach space X such that GL(X) is not connected. (By Theorem 5.1.6 such Banach spaces exist). Let GLI (E) be the connected component of GL(E) which contains the unit operator I. Take any operator A ∈ GL(X) \ GLI (X) and define I if z ∈ V1 , F (z) = A if z ∈ V2 . We interpret F as a ({U1 , U2 }, OGL(X) )-cocycle, setting F12 = F and F21 = F −1 on U1 ∩ U2 , F11 = I on U1 and F22 = I on U2 . Then F is not C GL(X) -trivial. Indeed, assume it is C GL(X) -trivial, i.e., F = C1 C2−1 on U1 ∩ U2 for certain continuous functions C1 : U1 → GL(X) and C2 : U2 → GL(X). Take r < ρ < R and set γ(t) = C1 ρe−it C2−1 ρeit for 0 ≤ t ≤ π. Then γ is a continuous curve in GL(X) connecting I and A, which is a contradiction.

5.7. Refinement of the covering

133

On the positive side, there is the following theorem, which is the main result of the present chapter: 5.6.3 Theorem. Let D ⊆ C be an arbitrary open set, let U be an arbitrary open covering of D, and let f be a (U, OG )-cocycle. Assume that at least one of the following conditions is satisfied: (i) D is simply connected. (ii) G is connected. (iii) f is C G -trivial. Then f is OG -trivial. We point out again the special case of a covering by two open sets, which is sufficient for many applications: 5.6.4 Corollary. Let D1 , D2 ⊆ C be two open sets, and let f : D1 ∩ D2 → G be holomorphic. Assume that at least one of the following conditions is satisfied: (i) D is simply connected. (ii) G is connected. (iii) There exist continuous functions cj : Dj → G with f = c1 c−1 2 on D1 ∩ D2 . Then there exist holomorphic functions fj : Dj → G with f = f1 f2−1 on D1 ∩ D2 . Note that by well-known topological results, each of the conditions (i) and (ii) in Theorem 5.6.3 implies condition (iii), but we do not use this topological fact. We prove directly that each of the conditions (i), (ii) or (iii) yields OG -triviality of f . In the case of condition (i) this will be done in Section 5.9, and in the case of conditions (ii) and (iii) we prove this in Section 5.11. Recall that, by Theorem 5.1.5, condition (ii) in Theorem 5.6.3 is satisfied if G = GL(H) where H is an Hilbert space. Therefore we have 5.6.5 Corollary (to Theorem 5.6.3). For each Hilbert space H and each open set D ⊆ C, any OGL(H) -cocycle over D is OGL(H) -trivial. In particular, for each n ∈ N∗ and each open set D ⊆ C, any OGL(n,C) -cocycle over D is OGL(n,C) -trivial.

5.7

Refinement of the covering

In this section we develop a technique which allows us to compare cocycles with different coverings. Throughout this section, A is a Banach algebra with unit 1, G a subgroup of the group of invertible elements of A, and D ⊆ C is an arbitrary open set.


134

5.7.1 Definition. Let U = {Uj }j∈I and V = {Vμ }μ∈J be two open coverings of D such that V is a refinement of U. Then (by definition of a refinement) there is a map τ : J → I with Vμ ⊆ Uτ (μ) for all μ ∈ J. For any such map τ and each cocycle f ∈ Z 1 (U, C G ) (Def. 5.6.1), we define a cocycle τ ∗ f ∈ Z 1 (V, C G ) setting (τ ∗ f )μν = fτ (μ)τ (ν) , Vμ ∩Vν

for all μ, ν ∈ J with Vμ ∩ Vν = ∅. We shall say that a cocycle g ∈ Z 1 (V, C G ) is induced by a cocycle f ∈ 1 Z (U, C G ) if there exists a map τ : J → I with Vμ ⊆ Uτ (μ) , μ ∈ J, such that g = τ ∗f . Note that in this case g ∈ Z 1 (V, OG ) if f ∈ Z 1 (U, OG ). Note also that, in general, for a cocycle f ∈ Z 1 (U, OG ), there exist different cocycles in Z 1 (V, C G ), which are induced by f . However, there is the following 5.7.2 Proposition. Let D ⊆ C be an open set, let U = {Uj }j∈I and V = {Vν }ν∈J be two open coverings of D such that V is a refinement of U, and let F = C G or F = OG . Further, let f, g ∈ Z 1 (U, F) and f, g ∈ Z 1 (V, F) such that f is induced by f and g is induced by g. Then the following are equivalent: (i) f and g are F-equivalent. (ii) f and g are F-equivalent. In particular, the following are equivalent: (i ) f is F-trivial. (ii ) f is F-trivial. Proof. By hypothesis, we have some maps τ, ϕ : J → I with f = τ ∗ f and g = ϕ∗ g. We first assume that f and g are F-equivalent. Then there is a family of functions hj ∈ F(Uj ), j ∈ I, with h−1 j fjk hk = gjk

on Uj ∩ Uk ,

(5.7.1)

for all j, k ∈ I with Uj ∩ Uk = ∅. Then we define a family of functions hμ ∈ F(Vμ ), μ ∈ J, setting hμ = hτ (μ) gτ (μ),ϕ(μ) on Vμ , μ ∈ J. Then −1 −1 h−1 ν fνμ hμ = gτ (ν),ϕ(ν) hτ (ν) fτ (ν),τ (μ) hτ (μ) gτ (μ),ϕ(μ)

on Vμ ∩ Vν ,

for all μ, ν ∈ J with Vμ ∩ Vν = ∅. By (5.7.1) this implies −1 h−1 ν fνμ hμ = gτ (ν),ϕ(ν) gτ (ν),τ (μ) gτ (μ),ϕ(μ)

on Vν ∩ Vμ ,

5.7. Refinement of the covering

135

for all μ, ν ∈ J with Vμ ∩ Vν = ∅. Since g satisfies the cocycle condition this further implies νμ on Vν ∩ Vμ , h−1 ν fνμ hμ = gϕ(ν),ϕ(μ) = g for all μ, ν ∈ J with Vμ ∩ Vν = ∅, i.e., f and g are F-equivalent. Now we assume that f and g are F-equivalent. Then there exists a family of functions hμ ∈ F(Vμ ), μ ∈ J, with h−1 νμ ν fνμ hμ = g

on Vμ ∩ Vν ,

for all μ, ν ∈ J, with Vμ ∩ Vν = ∅. Since f = τ ∗ f and g = ϕ∗ g, then h−1 ν fτ (ν),τ (μ) hμ = gϕ(ν),ϕ(μ)

on Vν ∩ Vμ ,

for all μ, ν ∈ J, with Vμ ∩ Vν = ∅. Since f and g satisfy the cocycle condition, this implies on Vν ∩ Vμ ∩ Uj , h−1 ν fτ (ν),j fj,τ (μ) hμ = gϕ(ν),j gj,ϕ(μ) and further, fj,τ (μ) hμ gϕ(μ),j = fj,τ (ν) hν gϕ(ν),j

on Vν ∩ Vμ ∩ Uj ,

for all j ∈ I and ν, μ ∈ J with Vν ∩ Vμ ∩ Uj = ∅. Therefore we can define a family of functions hj ∈ F(Uj ), j ∈ I, by setting hj = fj,τ (μ) hμ gϕ(μ),j

on Vμ ∩ Uj ,

for all j ∈ I and μ ∈ J with Vμ ∩ Uj = ∅. Using again that f and g satisfy the cocycle condition, we obtain −1 h−1 j fjk hk = gj,ϕ(μ) hμ fτ (μ),j fjk fk,τ (μ) hμ gϕ(μ),k = gjk

on Vμ ∩ Uj ∩ Uk ,

for all j, k ∈ I and μ ∈ J with Vμ ∩Uj ∩Uk = ∅. Hence f and g are F-equivalent. 5.7.3 Definition. Let D ⊆ C be an open set. By a C G -cocycle over D we mean a (U, C G )-cocycle such that U is an open covering of D. Correspondingly, by an OG -cocycle over D we mean a (U, OG )-cocycle such that U is an open covering of D. The open covering U then will be called the covering of this cocycle. In Definition 5.6.1 we introduced the notion of equivalence for cocycles with the same covering. In view of Proposition 5.7.2, the following definition is correct. 5.7.4 Definition. Let D ⊆ C be an open set. (i) Two C G -cocycles f and g over D will be called C G -equivalent or continuously equivalent over D if there exists an open covering W of D which is a refinement both of the covering of f and of the covering of g such that at least one (W, C G )-cocycle induced by f is C G -equivalent to at least one (W, C G )cocycle induced by g (or, what is the same (by Proposition 5.7.2), such that any (W, C G )-cocycle induced by f is C G -equivalent to any (W, C G )-cocycle induced by g).

136


(ii) Two OG -cocycles f and g over D will be called OG -equivalent or holomorphically equivalent over D if there exists an open covering W of D which is a refinement both of the covering of f and of the covering of g such that at least one (W, OG )-cocycle induced by f is OG -equivalent to at least one (W, OG )-cocycle induced by g. 5.7.5 Definition. Let D ⊆ C an open set, let D, and let Y be an open subset of D. Set U ∩ Y = Uj ∩ Y

U = {Uj }j∈I be an open covering of j∈I ,

and let F = C G or F = OG . Then we define: (i) Let f be an (U, F)-cocycle over D. Then we denote by f Y the (U ∩ Y, F)-cocycle defined by (f |Y )jk = fjk Uj ∩U

k ∩Y

for all j, k ∈ I with Uj ∩ Uk ∩ Y = ∅. This cocycle f Y will be called the restriction of f to Y . We shall say that f is F-trivial over Y if f Y is F-trivial. (ii) Let f, g be two F)-cocycles over D. Then we shall say that f and g are F-equivalent over Y if the restricted cocycles f Y and g Y are F-equivalent. 5.7.6 Proposition. Let D ⊆ C be an open set, let U = {Uj }j∈I be an open covering of D, let F = C G or F = OG , and let f be an F-cocycle over D, which is F-trivial over each Uj . Then f is F-equivalent to some (U, F)-cocycle. Proof. Let V be the covering of f . For each j ∈ I, we take an open covering Wj = {Wjν }ν∈N of Uj so fine that W := {Wjν) }(j,ν)∈I×N is a refinement of V. Let f = {fjμ,kν }(j,μ),(k,ν)∈I×N be a (W, F)-cocycle induced by f . Then f is Fequivalent to f (by Definition 5.7.4), and therefore it is sufficient to prove that f is F-equivalent to some (U, F)-cocycle. Since f is F-equivalent to f , each f |Uj is F-equivalent to f |Uj . Since each f |Uj is F-trivial, it follows that also each f |Uj is F-trivial. Hence, for all j ∈ I, we have a family hjν ∈ F (Wjν ), ν ∈ N, such that, for each j ∈ I, h−1 jν fjν,jμ hjμ = 1

on Wjν ∩ Wjμ

(5.7.2)

for all ν, μ ∈ N with Wjν ∩ Wjμ = ∅. Setting = h−1 fjν,kμ jν fjν,kμ hkμ

on Wjν ∩ Wkμ ,

for all (j, ν), (k, μ) ∈ I × N with Wjν ∩ Wkμ = ∅, we define a (W, F)-cocycle f . By its definition, f is F-equivalent to f . Hence f is F-equivalent to f . Since f

5.8. Exhausting by compact sets

137

satisfies the cocycle condition, we obtain that = h−1 fjν,kμ jν fjν,jμ fjμ ,kμ fkμ ,kμ hkμ

−1 −1 = h−1 jν fjν,jμ hjμ hjμ fjμ ,kμ hkμ hkμ fkμ ,kμ hkμ

on Wjν ∩ Wkμ ∩ Wjν ∩ Wkμ for all j, k ∈ I and ν, μ, ν , μ ∈ N with Wjν ∩ Wkμ ∩ Wjν ∩ Wkμ = ∅. By (5.7.2) this implies fjν,kμ = h−1 jμ fjμ ,kμ hkμ = fjν ,kμ

on Wjν ∩ Wkμ ∩ Wjν ∩ Wkμ

for all j, k ∈ I and ν, μ, ν , μ ∈ N with Wjν ∩ Wkμ ∩ Wjν ∩ Wkμ = ∅. Hence there is a well-defined family of functions fjk ∈ F(Uj ∩ Uk ), j, k ∈ I, defined by fjk = fjν,kμ

on Wjν ∩ Wkμ

for all j, k ∈ I and μ, ν ∈ N with Wjν ∩ Wkμ = ∅. Since f satisfies the cocycle condition, this implies fjk fkl = fjν,kμ fkμ,lλ = fjν,lλ

on Wjν ∩ Wkμ ∩ Wlλ

for all j, k, l ∈ I and ν, μ, λ ∈ N with Wjν ∩ Wkμ ∩ Wlλ = ∅. Hence fkl = fjl fjk

on Uj ∩ Uk ∩ Ul

for all j, k, l ∈ I with Uj ∩ Uk ∩ Wl = ∅, i.e., f ∈ Z 1 (U, F). Let τ : I × N → I be the map defined by τ (j, μ) = j for (j, μ) ∈ I × N. Since Wjν ⊆ Uj = Uτ (j,ν) and since, by definition of f , = fjk = fτ(jν)τ (kμ) fjν,kμ Wjν ∩Wkμ

for all (j, ν) ∈ I × N

Wjν ∩Wkμ

,

(j, ν), (k, μ) ∈ I × N,

then f is induced by f . Since f is F-equivalent to f , this implies that f is F-equivalent to f .

5.8

Exhausting by compact sets

Here we prove the following technical lemma which we shall use several times in this book:


138

5.8.1 Lemma. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A. Let D ⊆ C be an open set (possibly unbounded), let f be an OG -cocycle over D, and let (Dn )n∈N be a sequence of bounded open sets Dn ⊆ D such that: (1) Dn ⊆ Dn+1 for all n ∈ N. (2) n∈N Dn = D. (3) For each n ∈ N, any function from OG (Dn+1 ) can be approximated uniformly on Dn by functions from OG (D). (4) The cocycle f is OG -trivial over each Dn , n ∈ N. Then f is OG -trivial over D. Proof. Let · be the norm of A and set dist(a, A \ G) = inf a − b b∈A\G

for a ∈ G.

Let U = {Uj }j∈I be the covering associated to f . By Proposition 5.7.2, after passing to a refinement, we may assume that each Uj is a relatively compact open disc in D and fjk ∈ OG (U j ∩ U k ) for all j, k ∈ I. Note that then, for each j ∈ I, there exists nj ∈ N with U j ⊆ Dn if n ≥ nj . (5.8.1) Moreover we may assume that for each compact set K ⊆ D there exists only a (5.8.2) finite number of indices j ∈ I with Uj ∩ K = ∅. To prove the lemma it is sufficient to find a sequence fn n∈N of families fn = fnj j∈I of functions fnj ∈ OG (Dn+1 ∩ U j ) as well as a sequence (εn )n∈N of positive numbers, such that, for all n ∈ N, −1 fjk = fnj fnk

εn < max

z∈D n ∩U j

1 4

min

z∈D n ∩U j

on Dn+1 ∩ U j ∩ U k ,

j, k ∈ I,

(5.8.3)

dist fnj (z), A \ G ,

j ∈ I,

(5.8.4)

fnj (z) − fn−1,j (z) < εn−1 if n ≥ 1 ,

j ∈ I, and

(5.8.5)

εn−1 if n ≥ 1 . (5.8.6) 2 Indeed, then it follows from (5.8.1), (5.8.5) and (5.8.6) that, for all j ∈ I and n, m ∈ N with nj ≤ n < m, εn
0 such that

1 ε0 < dist f0,j (z), G \ A for all j ∈ I. min 4 z∈D0 ∩Uj With this choice of the family {f0j }j∈I and the number ε0 conditions (5.8.3)– (5.8.6) are satisfied for n = 0. Hypothesis of induction: Assume, for some m ∈ N, we already have families f0 = {f0j }j∈I , . . . , fm = {fmj }j∈I of functions f0j ∈ OG (D0 ∩ U j ), . . . , fmj ∈ OG (Dm ∩ U j )


140

as well as positive numbers ε0 , . . . , εm such that (5.8.5) -(5.8.4) hold for n = 0, . . . , m. Step of induction: Since the compact set Dm ∩ U j is contained in Dm+1 ∩ U j and fmj is continuous on Dm+1 ∩ U j , the function fmj is bounded on Dm ∩ U j . By condition (5.8.2), this implies that max

max

j∈I z∈D m ∩U j

By (5.8.7),

−1 fjk = fmj fmk

fmj (z) < ∞.

(5.8.9)

on Dm+2 ∩ Uj ∩ Uk .

Moreover, by hypothesis of induction, −1 fjk = fmj fmk


Since Dm+1 ⊆ Dm+2 , this yields −1 −1 = fmj fm,j fmk fmk


Hence, there is a well-defined function Φ ∈ OG (Dm+1 ) with −1 Φ = fmj fmj

(5.8.10)

on Dm+1 ∩ Uj for all j ∈ I. Note that, since fmj is continuous on Dm+1 ∩ U j and, by (5.8.8), fmj is continuous on Dm+2 ∩ U j , (5.8.10) even holds on Dm+1 ∩ U j , j ∈ I. By hypothesis of the lemma, Φ can be approximated uniformly on Dm by functions from OG (D). Therefore and by (5.8.9), we can find Ψ ∈ OG (D) such that εm max ΨΦ − 1 < for all j ∈ I. max fmj Dm D m ∩U j

Since (5.8.10) holds over Dm+1 ∩ U j and Dm ∩ U j ⊆ Dm+1 ∩ U j , this implies that −1 max Ψfmj fmj − 1
0 so small that condition (5.8.4) is satisfied for n = m + 1. As εm > 0, we may moreover assume that (5.8.6) holds for n = p + 1. From (5.8.7) we get −1 −1 −1 −1 fm+1,j fm+1,k = fmj Ψ Ψfmk = fmj fmk = fjk

5.9. The case of simply connected open sets

141

on Dm+2 ∩ U j ∩ U k , i.e., (5.8.3) holds for n = m + 1. From (5.8.11) it follows that

−1 − 1 fmj < εm . max fm+1,j − fmj = max Ψfmj fmj D m ∩U j

D m ∩U j

Hence also (5.8.5) holds for n = m + 1.

5.9

Proof of Theorem 5.6.3 for simply connected open sets

In this section A is a Banach algebra with unit 1, and G is an open subgroup of the group of invertible elements of A. The first step in the proof of Theorem 5.6.3 is the following 5.9.1 Lemma (Cartan lemma. Second version). Let (D1 , D2 ) be a Cartan pair such that C \ D1 ∪ D2 is connected, and let f ∈ OG (D1 ∩ D2 ) such that all values of f belong to the same connected component of G. Then there exist fj ∈ OG (Dj ), j = 1, 2, such that on D1 ∩ D2 . (5.9.1) f = f1−1 f2 Proof. Take a sequence of Cartan pairs (Dn,1 , Dn,2 ) n∈N such that • Dn,j ⊆ Dn+1,j ⊆ Dj for j = 1, 2 and all n ∈ N, • n∈N Dn,j = Dj for j = 1, 2 and • For each n ∈ N, C \ (D1,n ∪ D2,n ) is connected. Set D = D1 ∪D2 and Dn = Dn,1 ∪Dn,2 . Then, it follows from part (i) of the Runge approximation Theorem 5.0.1 (which we already proved at the end of Section 5.4) that the functions from OG (Dn+1 ) can be approximated uniformly on Dn by functions from OG (D), i.e., considered in Lemma 5.8.1. we have theGsituation Now we consider the {D1 , D2 }, O -cocycle F defined by F12 = f . We have to prove that F is OG -trivial. By the Cartan Lemma 5.5.2, F is OG -trivial over each Dn . Therefore, by Lemma 5.8.1, F is OG -trivial. Using this lemma and propositions 5.7.2 and 5.7.6, now we can prove the following special case of Theorem 5.6.3: 5.9.2 Lemma. Let K := z ∈ C 0 < Re z < 1 and 0 < Im z < 1 , and let Ω an open neighborhood of K. Then each OG -cocycle over Ω is OG -trivial over K.


142

Proof. Let an OG -cocycle f over Ω be given, let V = {Vν }ν∈I be the covering of f , and let V ∩ K := {Vν ∩ K}ν∈I . We choose n ∈ N sufficiently large and denote by Ujk , j, k = 1, . . . , n, the open rectangle of all z ∈ K with 1 1 1 1 k−1− < Re z < k + 3 n 3 n and

j−1−

1 3

1 < Im z < n

j+

1 3

1 . n

Then U := {Ujk }1≤j,k≤n is a refinement of V ∩ K. Let f be a (U, O)-cocycle induced by f . By Proposition 5.7.2, it is sufficient to prove that f is OG -trivial. To do this we give the family U an order saying that Ujk < Uj k , if and only if, either j < j or j = j and k < k . Let U1 , . . . , Un2 be the family U numbered in this way. Now we prove by induction that, for all 1 ≤ j ≤ N 2 , the cocycle f is OG trivial over U1 ∪ . . . ∪ Uj . (For j = n2 this is the assertion which we have to prove.) Since U1 belongs to the covering associated to f , it is clear that f is OG trivial over U1 . Assume, for some 1 ≤ j ≤ N 2 − 1, we already know that f is OG -trivial over U1 ∪ . . . ∪ Uj . As f is also OG -trivial over Uj+1 , then it follows from Proposition 5.7.6 that f is OG -equivalent to some {U1 ∪ . . . ∪ Uj , Uj+1 }, OG -cocycle f . Since, clearly, (U1 ∪ . . . ∪ Uj , Uj+1 ) is a Cartan pair (Definition 5.5.1) such that C \ (U1 ∪ . . . ∪ Uj ) ∩ Uj+1 is connected, it follows from Lemma 5.9.1 that f is OG -trivial. Hence f is OG -trivial. Proof of Theorem 5.6.3 if D is simply connected. Since D is simply connected, by the Riemann mapping theorem we may assume that either D = C or D is the open unit disc. In both cases, there exists a sequence (Dn )n∈N of open discs Dn ⊆ C such that: • Dn ⊆ D, • Dn ⊆ Dn+1 for all n ∈ N, • n∈N Dn = D, • by part (i) of the Runge approximation Theorem 5.0.1 (which we already proved at the end of Section 5.4), for all n ∈ N, any f ∈ OG (Dn+1 ) can be approximated uniformly on Dn by functions from OG (D). Hence, by Lemma 5.8.1, it is sufficient to prove that any OG -cocycle over D becomes OG -trivial after restriction to each Dn , which is indeed the case by Lemma 5.9.2 (again using the Riemann mapping theorem). Finally we point out the following special case of Theorem 5.6.3, which is now proved:

5.10. Runge approximation of G-valued functions

143

5.9.3 Corollary. Let D1 , D2 ⊆ C be two open sets such that D1 ∪ D2 is simply connected, and let f : D1 ∩ D2 → G be holomorphic. Then there exist holomophic functions fj : Dj → G, j = 1, 2, such that f = f1−1 f2 on D1 ∩ D2 . In particular, this proves the assertion of Theorem 0.0.1 if D1 ∪ D2 is simply connected.

5.10

Runge approximation of G-valued functions General case

The aim of this section is to prove part (ii) of the Runge approximation Theorem 5.0.1. Recall that part (i) of Theorem 5.0.1 was proved already at the end of Section 5.4. Then, using part (i) of Theorem 5.0.1, at the end of the preceding section we obtained Corollary 5.9.3, which now, in the present section, will be combined with part (i) of Theorem 5.0.1 to prove part (ii) of Theorem 5.0.1. Doing this, it is convenient to pass to the Riemann sphere and to prove the slightly more general Runge approximation Theorem 5.10.5 below.6 First let us shortly recall the notion of the Riemann sphere. The Riemann sphere is given by the set P1 := C ∪ {∞} endowed with a topology and a notion of holomorphic functions what we now are going to explain: Definition of the topology: A subset U of P1 is called open if either ∞ ∈ U and U is open as a subset of C (with respect to the usual Euclidean topology of C) or ∞ ∈ U and C \ U is compact (again with respect to the Euclidean topology of C). It is easy to see that in this way P1 becomes a topological space, which is, by stereographic projection7 , homeomorphic to the 2-dimensional sphere. Definition of holomorphic functions: Set 1 =∞, 0

1 =0 ∞

and

z ± ∞ = ∞ ± z = ∞ for all z ∈ C.

Then, for each a ∈ P1 , a map Ta : P1 → P1 is well-defined, by setting 1/(z − a) if a = ∞, Ta (z) = z if a = ∞. 6 Theorem 5.0.1 is the special case ∞ ∈ D of Theorem 5.10.5 where part (i) corresponds to the case of connected P1 \ D and part (ii) corresponds to the case of non-connected P1 \ D. 7 Let us identify the complex plane C with the plane in R3 which consists of the points of the form (x, y, 0), x, y ∈ R, and let S be the sphere of radius 1 centered at (0, 0, 1) ∈ R3 . Let N := (0, 0, 2) ∈ S be the ”north pole” of S. For each p ∈ S \ {N }, we denote by L(p) the line through p and N . Then L(p) intersects C in precisely one point. Let us denote this point by C ∩ L(p). The bijective map π : S → P1 defined by C ∩ L(p) if p ∈ S \ {N }, π(p) = ∞ if p = ∞,

is called the stereographic projector.

144


It is easy to see that, for each a ∈ P1 , Ta is a homeomorphism of P1 with Ta (a) = ∞ and Ta (P1 \ {a}) = C. If there is no danger of confusion, we use also the following notation: • The letter z denotes the identical map T∞ . • If a ∈ P1 and a = ∞, then the expression

1 z−a

denotes the map Ta .

Now let U ⊆ P1 be open, U = ∅, and let f : U → C be a function. If ∞ ∈ U , i.e., U ⊆ C, then we already know what it means that f is holomorphic on U . If ∞ ∈ U , then we define: f is holomorphic on U if the following two conditions are fulfilled: • f is holomorphic on U \ {∞}. • f ◦ T0 = f z1 is holomorphic on T0−1 (U ). Note that, by Riemann’s removability theorem, this is the case, if and only if, f is continuous on U , and the restriction of f to U \ {∞} is holomorphic. The ring of all holomorphic functions defined on U will be again denoted by O(U ). We summarize: The Riemann sphere P1 is given by the triplet C ∪ {∞} , T , O where T is the family of open subsets of C ∪ {∞} defined above, and O is the map T \ {∅} U → O(U ). Now let E be a Banach space, and let U ⊆ P1 be an open set, U = ∅. If ∞ ∈ U , i.e., U ⊆ C, then we already defined what a holomorphic E-valued function is. If ∞ ∈ U , then, as in the case of scalar functions, we define: f is holomorphic on U , if and only if, • f is holomorphic on U \ {∞}. • f z1 is holomorphic on T0−1 (U ). The space of all holomorphic E-valued functions on U will be again denoted by OE (U ). Now let D ⊂ P1 be a non-empty open set, and let f be a map from D to 1 P . If ∞ ∈ f (D), i.e., f (D) ⊆ C, then we just defined what it means that f is holomorphic. If ∞ ∈ f (D), then we define: f is holomorphic on D if the restriction of f to D \ f −1 (∞) is holomorphic and, moreover, the function 1 f ◦ Tf −1 (∞) = f z − f −1 (∞) is holomorphic on Tf−1 −1 (∞) . If D and D are two open sets in C, then we say that f is biholomorphic from D onto D , if and only if, f is bijective from D onto D such that f is holomorphic on D and f −1 is holomorphic on D . Note that, for each a ∈ P1 , Ta is biholomorphic from P1 onto P1 where P1 \{a} is mapped onto C.

5.10. Runge approximation of G-valued functions

145

An open set D ⊆ P1 will be called an open set with piecewise C 1 -boundary if D = P1 and, for a ∈ P1 \ D, the open set Ta (D) (which is contained in C) is an open set with piecewise C 1 -boundary as defined in Section 1.4.1. Since, for each a ∈ P1 , Ta is biholomorphic between P1 \ {a} and the complex plane, from the theory of holomorphic functions in the complex plane we get a corresponding theory on each set of the form P1 \ {a} where a is an arbitrary point in P1 . For example, from part (i) of the Runge approximation Theorem 5.0.1 (which is already proved) we immediately get: 5.10.1 Proposition. Let D be an open set with piecewise C 1 boundary in P1 such that P1 \ D is connected, let G be an open subgroup of the group of invertible elements of a Banach algebra with unit, and let f : D → G be a continuous function which is holomorphic in D such that all values of f belong to the same connected component of G. Then, for each a ∈ P1 \ D, f can be approximated uniformly on D by functions from OG (P1 \ {a}). Moreover, from Corollary 5.9.3 we immediately get: 5.10.2 Proposition. Let D1 and D2 be two open sets in P1 such that P1 \ (D1 ∪ D2 ) is connected and not empty. Then, for each holomorphic function f : D1 ∩D2 → G there exist holomorphic functions fj : Dj → G, j = 1, 2, such that f = f1−1 f2 on D1 ∩ D2 . 5.10.3 Definition. If X is a subset of P1 and G an open subgroup of the group of invertible elements of G, then we denote by OG (X) the group of continuous G-valued functions on X which are holomorphic in the interior of X. 5.10.4 Lemma. Let D ⊆ P1 be an open set with piecewise C 1 -boundary, and let U1 , . . . , Un be the connected components of P1 \ D. Let n ≥ 2 and let some points aj ∈ Uj , 1 ≤ j ≤ n, be chosen. Further, let G be an open subgroup of the group of invertible elements of a Banach algebra with unit, and let f ∈ OG (D). Then there exist functions fj ∈ OG (P1 \ Uj ), 1 ≤ j ≤ n, and a function h ∈ OG P1 \ {a1 , . . . , an } such that f = hfn . . . f1 on D. Proof. For 1 ≤ k ≤ n, we consider the following statement: A(k): There exist functions fj ∈ OG P1 \ Uj , 1 ≤ j ≤ k, and a function hk ∈ OG D ∪ U1 \ {a1 } ∪ . . . ∪ Uk \ {ak } such that f = hk fk . . . f1 on D. Since

D ∪ U1 \ {a1 } ∪ . . . ∪ Un \ {an } = P1 \ {a1 , . . . , an },

then A(n) is the assertion of the lemma. Therefore it is sufficient to prove A(1) and the conclusions A(k) ⇒ A(k + 1), 1 ≤ k ≤ n − 1. Proof of A(1): Since 1 P \ U 1 ∪ D ∪ (U 1 \ {a1 }) = P1 \ {a1 }


146 and

P1 \ U 1 ∩ D ∪ (U 1 \ {a1 }) = D, from Proposition 5.10.2 we get functions f1 ∈ OG P1 \ U 1 and h1 ∈ OG D ∪ U 1 \ {a1 } such that (5.10.1) f = h1 f1

on D. Since f is continuous and with values in G on D, since h1 is continuous and with values in G on D ∪ ∂U1 , since f1 is continuous and with values in G on D \ ∂U holds in D, it follows that f1 ∈ OG (P1 \ U1 ), 1 and since (5.10.1) G h1 ∈ O D ∪ (U1 \ {a1 }) and (5.10.1) holds on D, i.e., assertion A(1) is valid. Proof of A(k) ⇒ A(k +1): Let 1 ≤ k ≤ n−1 be given, assume that statement A(k) is valid, and let f1 , . . . , fk and hk be as in this statement. Since

P1 \ U k+1 ∪ D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } = P1 \ {ak+1 } and

P1 \ U k+1 ∩ D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } = D ∪ U 1 \ {a1 } ∪ . . . ∪ U k \ {ak } , from Proposition 5.10.2 we get functions

G fk+1 ∈ OZ,m P1 \ U k+1 G D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } hk+1 ∈ OZ,m

and such that

hk = hk+1 fk+1 (5.10.2) on D ∪ U 1 \ {a1 } ∪ . . . ∪ U k \ {ak } . Since hk is continuous and with values in G on D, since hk+1 is continuous and with values in G on D ∪ ∂Uk+1 , since fk+1 is continuous and with values in G on D \ ∂Uk+1 and since (5.10.2) holds in D, it follows that

1 fk+1 ∈ OG Z,m P \ Uk+1 , hk+1 ∈ OG Z,m D ∪ U1 \ {a1 } ∪ . . . ∪ Uk+1 \ {ak+1 }

and (5.10.2) holds on D. Since f = hk fk . . . f1 on D, this implies that f = hk+1 fk+1 fk . . . f1 on D, i.e., assertion A(k + 1) is valid.

5.11. Proof of the main theorem in the general case

147

5.10.5 Theorem (Runge approximation). Let G be an open subgroup of the group of invertible elements of a Banach algebra with unit. Let D ⊆ P1 be an open set with piecewise C 1 -boundary, and let f : D → G be a continuous function which is holomorphic in D such that all values of f belong to the same connected component of G. Further, let U1 , . . . , Un be the connected components of P1 \ D, and let some points aj ∈ Uj , 1 ≤ j ≤ n, be given. Then f can be approximated uniformly on D by functions from OG P1 \ {a1 , . . . , an }). Proof. If n = 1, the assertion of the theorem is that of Proposition 5.10.1. If n ≥ 2, then, by Lemma 5.10.4, f can be written in the form f = hfn . . . f1

on D, (5.10.3) where fj ∈ OG (P1 \ Uj ), 1 ≤ j ≤ n, and h ∈ OG P1 \ {a1 , . . . , an } . Let V be the interior of P1 \ Uj . Since the boundary of Uj is piecewise C 1 (as a part of the boundary of D), also the boundary of V is piecewise C 1 and V = P1 \ Uj . Since Uj is connected (as a connected component of some set), P1 \ V = Uj is connected. Therefore, Proposition 5.10.1 can be applied to each Vj . Hence, each fj can be approximated uniformly on V = P1 \Uj by functions from OG P1 \{aj } . Since O P1 \ {aj }, G ⊆ OG P1 \ {p1 , . . . , pn } and D ⊆ P1 \ Uj ), this means in particular that each fj can be approximated uniformly on D by functions from OG P1 \ {p1 , . . . , pn } . Since h belongs to OG P1 \ {p1 , . . . , pn } and by (5.10.3), this implies the assertion of the theorem.

5.11

Proof of Theorem 5.6.3 in the general case

Here we prove Theorem 5.6.3. Recall that the sufficiency of condition (i) in Theorem 5.6.3 (that D is simply connected) was already proved at the end of Section 5.9. So it remains to prove that condition (ii) and condition (iii) in Theorem 5.6.3 are sufficient. Throughout this section, A is a Banach algebra with unit 1, GA is the group of invertible elements of A, G1 A is the connected component of the unit element in GA, and G is an open subgroup of GA. 5.11.1 Lemma. Let (D1 , D2 ) be a Cartan pair (Definition 5.5.1) such that D1 and D2 are connected, and let F ∈ OG (D1 ∩ D2 ) (Def. 5.4.5). Then the following are equivalent: (i) All values of F belong to the same connected component of G. (ii) There exist functions Fj ∈ OG (Dj ), j = 1, 2, such that F = F1−1 F2 on D1 ∩ D2 . (iii) There exist functions Cj ∈ C G (Dj ), j = 1, 2, such that F = C1−1 C2 on D1 ∩ D2 .


148

Proof. The conclusion (i)⇒(ii) is the assertion of the Cartan Lemma 5.5.2. The conclusion (ii)⇒(iii) is trivial. To prove (iii)⇒(i), we consider two arbitrary points z, w ∈ D1 ∩ D2 . Since D1 and D2 are connected, we can find continuous curves γj : [0, 1] → Dj , j = 1, 2, such that γj (0) = z and γj (1) = w, j = 1, 2. If now condition (iii) is satisfied, then, by setting γ(t) = C1−1 γ1 (t) C2 γ( t) , 0 ≤ t ≤ 1, we can define a continuous curve in G, where F (z) = C1−1 (z)C2 (z) = C1−1 γ1 (0) C2 γ2 (0) = γ(0) and

F (w) = C1−1 (w)C2 (w) = C1−1 γ1 (1) C2 γ2 (1) = γ(1),

i.e., γ connects F (z) and F (w).

5.11.2 Lemma. Let D ⊆ C be a bounded, connected open set with piecewise C boundary such that C \ D consists of n connected components, n ≥ 2. Then there exists a Cartan pair (D1 , D2 ) with D = D1 ∪ D2 and satisfying the following conditions: 1

(1) D1 is simply connected; (2) D2 is connected; (3) C \ D2 consists of n − 1 connected components. Proof. Let U1 , . . . , Un be the connected components of C \ D where U1 denotes the unbounded component. Let ∂Uj be the boundary of Uj . Choose points a1 ∈ ∂U1 and a2 ∈ ∂U2 which are smooth points of the boundary ∂D of D. Since D is connected and hence (as an open set) arcwise connected and since ∂D is piecewise C 1 , D is arcwise connected. Therefore we can find a continuous curve ϕ : [0, 1] → D with ϕ(0) = a1 and ϕ(1) = a2 . Moreover, since a1 and a2 are smooth points of ∂D, we can achieve that ϕ is a C ∞ -diffeomorphism between [0, 1] and ϕ([0, 1]) and ϕ([0, 1]) meets ∂D transversally (in both points, a1 and a2 ). Choose a neighborhood V in C of the interval [0, 1] so small that there exists an extension of ϕ to some C ∞ -diffeomorphism Φ from V onto some neighborhood W of γ([0, 1]). If δ > 0 and ε < ε , then we set K(δ; ε, ε ) = z = x + iy ∈ C − δ < x < 1 + δ and ε < y < ε . Choose δ0 > 0 and ε0 so small that K(δ0 ; −ε0 , ε0 ) ⊆ V . Since γ([0, 1]) meets ∂D transversally in the smooth points a1 and a2 and since D is connected, further then we can choose 0 < ε1 < ε0 so small that, for all −ε1 < ε < ε < ε1 , D1 (ε, ε ) := D ∩ Φ K(δ0 ; ε, ε ) and D2 (ε, ε ) := D \ D1 (ε, ε )

5.11. Proof of the main theorem in the general case

149

are connected bounded open sets with piecewise C 1 -boundary. Since U1 ∪ U2 ∪ γ([0, 1]) is connected, then U1 and U2 are contained in the same connected component of C \ D2 (ε, ε ) and the number of connected components of C \ D2 (ε, ε ) is n − 1. Moreover, choosing this ε1 sufficiently small, we can achieve that, for all −ε1 < ε < ε < ε1 , the open set Φ−1 D1 (ε, ε ) is star shaped with respect to point 1/2 and therefore simply connected. Hence, D1 (δ0 ; ε, ε ) is simply connected for all −ε1 < ε < ε < ε1 . We summarize: For all −ε1 < ε < ε < ε1 , D1 (ε, ε ) and D2 (ε, ε ) are connected bounded open sets with piecewise C 1 - boundary where the open set D1 (ε, ε ) is simply connected and C \ D2 (ε, ε) consists of n − 1 connected components. Now we fix some 0 < ε < ε1 /2 and set D1 := D1 (−2ε, 2ε)

and D2 := D2 (−ε, ε).

Then the intersection D1 ∩ D2 consists of the two connected components D1 (−2ε, −ε) and D1 (ε, 2ε) and (as mentioned in the summary above) each of these components is simply connected and has piecewise C 1 -boundary. Moreover, it is clear that D1 ∪ D2 = D and (D1 \ D2 ) ∩ (D2 \ D1 ) = ∅. Hence (D1 , D2 ) is a Cartan pair with D1 ∪ D2 . That this Cartan pair has the properties (1), (2) and (3) was also already mentioned in the above summary. 5.11.3 Lemma. Let D ⊆ C be an open set with piecewise C 1 -boundary, let U be a neighborhood of D and f a OG -cocycle over U . Suppose at least one of the following two conditions is fulfilled: (a) G is connected. (b) f is C G -trivial over U . Then f is OG -trivial over D. Proof. We may assume that D is connected. Further we proceed by induction over the number of connected components of C \ D. Beginning of induction: Suppose this number is 1, i.e., C \ D is connected. As the boundary of D is piecewise C 1 , then also C \ D is connected, which means that D is simply connected. Therefore the assertion of the lemma follows from the fact that the claim of Theorem 5.6.3 was already proved (at the end of Section 5.9) if condition (i) in Theorem 5.6.3 is satisfied. Hypothesis of induction: Assume, for some n ∈ N with n ≥ 2, the assertion of the lemma is already proved if the number of connected components of C \ D is n − 1. Step of induction: Assume that the number of connected components of C \ D is equal to n. Then, by Lemma 5.11.2, we can find a Cartan pair (D1 , D2 ) with


150

D = D1 ∪ D2 satisfying conditions (1), (2), (3) (of Lemma 5.11.2). Since the boundaries of D1 , D2 , D1 ∩ D2 and D are piecewise C 1 , we can find a Cartan pair (D1 , D2 ) satisfying the same conditions (1), (2), (3) such that Dj ⊆ Dj and

D1 ∪ D2 ⊆ U . Then f is OG -trivial over D1 , again by the fact that the claim of Theorem 5.6.3 was already proved (at the end of Section 5.9) if condition (i) in Theorem 5.6.3 is satisfied. Moreover, since the number of connected components of C \ D2 is G equal to n − 1, it follows from the hypothesis of induction that f is also O -trivial over D2 . By Proposition 5.7.6, this implies that f D ∪D is OG -equivalent to a 1

2

certain ({D1 , D2 }, OG )-cocycle f . Since D1 ∩ D2 ⊆ D1 ∩ D2 , setting F := f12 D ∩D 1

2

we obtain a function F ∈ OG (D1 ∩ D2 ). We claim that all values of F belong to the same connected component of G. If G is connected, this is trivial. If not, then condition (b) in the lemma under proof is satisfied, i.e., f is CG -trivial over U . As D1 ∪ D2 ⊆ U , then f is also C G trivial over D1 ∪D1 . Since f D ∪D is C G -equivalent to f (it is even OG -equivalent 1

2

to f ), this implies that also f is C G -trivial, i.e., we can find Cj ∈ C G (Dj ), j = 1, 2, with f12 = C1−1 C2 on D1 ∩ D2 .

Hence condition (iii) in Lemma 5.11.1 is satisfied, and it follows from Lemma 5.11.1 that all values of F belong to the same connected component of G. Since all values of F belong to the same connected component of G, it follows from the Cartan Lemma 5.5.2 (or from Lemma 5.11.1) that there exist functions Fj ∈ OG (Dj ), j = 1, 2, with F = F1−1 F2

on Dj ∩ Dj .

G Since F |D1 ∩D2 = f12 | D1 ∩D2 , this means in particular that f D is O -trivial. Finally, as f D ∪D and f are OG -equivalent and therefore f |D and f |D are 1

2

OG -equivalent, it follows that f is OG -trivial over D.

Proof of Theorem 5.6.3. Since the sufficiency of condition (i) in Theorem 5.6.3 (that D is simply connected) was already proved at the end of Section 5.9, we may assume that at least one of the following two conditions is fulfilled: (a) G is connected. (b) f is C G -trivial over U . bounded open sets with C 1 -boundaries such that Take a sequence (Ωn )n∈N of Ωn ⊆ Ωn+1 for all n ∈ N and n∈N Ωn = D. Let Un be the union of all bounded

5.12. OG

∞

(E)

-cocycles

151

connected components of C\Ωn which are subsets of D (if there are any – otherwise Un := ∅), and set Dn = Ωn ∪ U n . open sets with C 1 -boundaries such Then also (Dn )n∈N is a sequence of bounded that Dn ⊆ Dn+1 for all n ∈ N and n∈N Dn = D. Moreover this sequence has the important property that each bounded connected component of C \ Dn (if there are any) contains at least one point of C \ D. By the Runge approximation Theorem 5.0.1, this implies that, for each n, the functions from OG (Dn ) can be approximated uniformly on Dn by functions from OG (D). In particular we see that the sequence (Dn )n∈N has the properties (1), (2), (3) of Lemma 5.8.1. Therefore, by this lemma, for the OG -triviality of f it is sufficient that each f |Dn is OG -trivial, which is indeed the case, by Lemma 5.11.3.

5.12 OG

∞

(E)

-cocycles

In this section E is a Banach space. 5.12.1 Definition. (i) We denote by F ω (E) the ideal of all compact operators from L(E), and by F ∞ (E) we denote the ideal of all operators from F ω (E), which can be approximated by finite dimensional operators. (ii) Let G ω (E) be the group of invertible operators of the form I + K with K ∈ F ω (E), and let G ∞ (E) be the group of invertible operators of the form I + K with K ∈ F ∞ (E). (iii) We denote by FIω (E) the subalgebra of L(E) of operators of the form λI + K with K ∈ F ω (E) and λ ∈ C, and by FI∞ (E) we denote the subalgebra of L(E) of operators of the form λI + K with K ∈ F ∞ (E) and λ ∈ C. (iv) We set GFIω (E) = GL(E) ∩ FIω (E)

and GFI∞ (E) = GL(E) ∩ FI∞ (E).

In the following proposition we collect some well-known facts on these algebras and groups, which will be used without further reference throughout this book. 5.12.2 Proposition. The operators from F ∞ (E) are compact. Hence, zero is the only possible accumulation point of the spectrum of such an operator, and all other points of the spectrum are eigenvalues of finite multiplicity. Therefore it is easy to see that: (i) For dim E < ∞, the factor algebras FI∞ (E)/F ∞ (E) and FIω (E)/F ω (E) are isomorphic to C. (ii) If dim E = ∞ and λI + K = λ I + K where K, K ∈ F ω (E) and λ, λ ∈ C, then λ = λ and K = K .

152


(iii) The algebras FI∞ (E) and FIω (E) are closed in L(E) with respect to the operator norm. (iv) Let dim E = ∞, let D ⊆ C be an open set, and let f : D → FIℵ (E) be a holomorphic (continuous) function, where ℵ stands for one of the symbols ∞ or ω. Then f is of the form f (z) = λ(z) + K(z) ,

z ∈ D,

where λ : D → C and K : D → F ℵ (E) are uniquely determined holomorphic (continuous) functions. (v) The sets GFI∞ (E), G ∞ (E), GFIω (E) and G ω (E) are connected and closed subgroups of GL(E). The group GFI∞ (E) is the group of invertible elements of the Banach algebra FI∞ (E), and the group GFIω (E) is the group of invertible elements of the Banach algebra FIω (E). 5.12.3 Remark. It is easy to see that there is no Banach algebra which contains G ∞ (E) or G ω (E) as an open subgroup of its group of invertible elements. Therefore we cannot immediately apply to G ∞ (E) and G ω (E) the theory of cocycles as developed above. We circumvent this problem passing to the groups GFI∞ (E) and GFIω (E) , which are groups of invertible elements of a Banach algebra. Note however that G ∞ (E) and G ω (E) are complex Banach Lie groups and that the theory of Grauert and Bungart mentioned in the introduction to this book is developed for such groups. So, the theory of OG -cocycles, presented in this chapter, is valid for arbitrary Banach Lie groups. But, since G ∞ (E) and G ω (E) are the only examples of true Banach Lie groups, which we meet in this book, for simplicity, we avoid the notion of a general Banach Lie group. 5.12.4 Definition. Let D ⊆ C be an arbitrary open set, let U = {Uj }j∈I be an open covering of D, let A = {Ajk }j,k∈I be a U, OGL(E) -cocycle (Def. 5.6.1), and let ℵ stand for one of the symbols ∞ or ω. ℵ ℵ (i) The cocycle A will be called a U, OG (E) -cocycle or simply an OG (E) cocycle, if for all j, k ∈ I with Uj ∩ Uk = ∅, Ajk (z) ∈ G ℵ (E)

for all z ∈ Uj ∩ Uk .

ℵ

(ii) The cocycle A will be called OG (E) -trivial, if there exists a family of holomorphic functions Aj : Uj → G ℵ (E), j ∈ I, such that Ajk = Aj A−1 k

on Uj ∩ Uk

for all j, k ∈ I with Uj ∩ Uk = ∅. 5.12.5 Theorem. Let D ⊆ C be an open set, let U be an open covering of D, and ℵ let A be a U, OG (E) -cocycle, where ℵ stands for one of the symbols ∞ or ω. ℵ Then A is OG (E) -trivial.

5.13. Weierstrass theorems

153

Proof. Since G ℵ (E) is contained in GFIℵ (E), the cocycle A can be viewed as ℵ a U, OGFI (E) -cocycle. Since GFIℵ (E) is connected, it follows from Theorem ℵ 5.6.3 that A is OGFI (E) -trivial. Hence we have a family of holomorphic functions j : Uj → GF ℵ (E), j ∈ I, such that A I −1 j A Ajk = A k

on Uj ∩ Uk

(5.12.1)

for all j, k ∈ I with Uj ∩ Uk = ∅. For dim E < ∞ this completes the proof. Let dim E = ∞, and let λj : Uj → C∗ and Kj : Uj → F ℵ (E) be the holomorphic functions with j = λj I + Kj , A j ∈ I. Passing to the factor algebra L(E)/F ℵ (E), then it follows from (5.12.1) that λj = λk

on Uj ∩ Uk

for all j, k ∈ I with Uj ∩ Uk = ∅. Therefore it remains to set Aj = λ−1 j Aj .

We point out again the special case of Theorem 5.12.5 for coverings by two open sets, which is sufficient for many applications: 5.12.6 Corollary. Let D1 , D2 ⊆ C be two open sets, and let A : D1 ∩ D2 → G ℵ (E) be holomorphic, where ℵ stands for one of the symbols ∞ or ω. Then there exist holomorphic functions Aj : Dj → G ℵ (E) with A = A1 A−1 2 on D1 ∩ D2 .

5.13

Weierstrass theorems

Here we prove Theorem 0.0.2 stated in the introduction to this book. We do this in a more general setting of Banach algebras. Throughout this section, A is a Banach algebra with unit 1, and G is an open subgroup of the group of invertible elements of A. We prove: 5.13.1 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) D is simply connected. (ii) G is connected. Then there exist a holomorphic function h : D \ Z → G and holomorphic functions hw : Uw → G, w ∈ Z, such that hw fw = h

on Uw \ {w} .

(5.13.1)

The topological conditions (i) and (ii) in Theorem 5.13.1 can be replaced by the more general condition that the problem can be solved continuously, i.e., there is the following Oka-Grauert principle:


154

5.13.2 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Assume that: (iii) There exist a continuous function c : D \ Z → G and continuous functions cw : Uw → G, w ∈ Z, such that cw fw = c on Uw \ {w}, w ∈ Z. Then there exist a holomorphic function h : D \ Z → G and holomorphic functions hw : Uw → G, w ∈ Z, such that hw fw = h

on Uw \ {w} .

(5.13.2)

Proof of Theorems 5.13.1 and 5.13.2. We choose neighborhoods Vw ⊆ Uw , w ∈ Z, so small that Vw ∩ Vv = ∅ for all v, w ∈ Z with w = v. It is sufficient to find h ∈ OG (D \ Z) and hw ∈ OG (Vw ), w ∈ Z, such that hw fw = h

(5.13.3)

on Vw \ {w}. Indeed, since Vw ∩ Z = Uw ∩ Z = {w}, then, by (5.13.3), each hw admits an extension to a function from OG (Uw ), which we also denote by hw , such that (5.13.2) holds. Set D1 = w∈Z Vw and D2 = D \ Z. Since the sets Vw are pairwise disjoint and Vw ∩ Z = {w}, the family of functions fw can be interpreted as a single holomorphic function f ∈ OG (D1 \ Z) = OG (D1 ∩ D2 ). Now, by Corollary 5.6.4, there exist hj ∈ OG (Dj ), j = 1, 2, with f = h−1 1 h2 on D1 ∩D2 . Setting hw = h1 Vw and h = h2 , we complete the proof. Since, in theorems 5.13.1 and 5.13.2, the multiplication by the functions hw is carried out from the left, we call these theorems left-sided Weierstrass theorems. There are also right- and two-sided versions. If the multiplication in A is denoted by “·”, then we can pass to the Banach which consists of the same additive group A but with the multiplication Algebra A “ · ” defined by a · b = b · a. In this way, from theorems 5.13.1 and 5.13.2 we get the following right-sided Weierstrass theorems: 5.13.3 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) D is simply connected. (ii) G is connected. Then there exist a holomorphic function h : D \ Z → G and holomorphic functions hw : Uw → G, w ∈ Z, such that fw hw = h

on Uw \ {w} .

(5.13.4)

5.13. Weierstrass theorems

155

5.13.4 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Assume that: (iii) There exist a continuous function c : D \ Z → G and continuous functions cw : Uw → G, w ∈ Z, such that fw cw = c on Uw \ {w}, w ∈ Z. Then there exist a holomorphic function h : D \ Z → G and holomorphic functions hw : Uw → G, w ∈ Z, such that fw hw = h

on Uw \ {w} .

(5.13.5)

Finally, we present a two-sided Weierstrass theorem: 5.13.5 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and two holomorphic functions fw , gw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) D is simply connected. (ii) G is connected. Then there exist a holomorphic function h : D \ Z → G and holomorphic functions hw : Uw → G, w ∈ Z, such that fw hw gw = h

on Uw \ {w} .

(5.13.6)

Proof. Let a family of positive integers mw , w ∈ Z, be given. Then from the leftsided Weierstrass Theorem 10.1.1 we get a holomorphic function hl : D \ Z → G and holomorphic functions hlw : Uw → G, w ∈ Z, such that hlw gw = hl

on Uw \ {w}.

(5.13.7)

From the right-sided Weierstrass Theorem 10.2.1 we get a holomorphic function hr : D \ Z → G and holomorphic functions hrw : Uw → G, w ∈ Z, such that fw hrw = hr

on Uw \ {w}.

(5.13.8)

Set h = hl hr and hw = hrw hlw , w ∈ Z. Then h ∈ OG (D \ Z), hw ∈ OG (Uw ) and fw hw gw = fw hrw hlw gw = hr hl = h

on Uw \ {w}.

5.13.6 Remark. Instead of conditions (i) or (ii) in Theorem 5.13.5 also the following condition would be sufficient (Oka-Grauert principle): (iii) There exist a continuous function c : D \ Z → G and continuous functions cw : Uw → G, w ∈ Z, such that fw cw gw = c on Uw \ {w}, w ∈ Z.

156


But to prove this we would need a generalization of the theory of multiplicative cocycles where the group G in Definition 9.1.2 is replaced by a fiber bundle of groups with characteristic fiber G. This generalization is well known in Complex analysis of several variables, as a part of the theory of Grauert [Gr1, Gr2, Gr3] and Bungart [Bu] mentioned in the introduction to the present book. To keep this book simpler and shorter we omit this.

5.14

Weierstrass theorems for G ∞ (E) and G ω (E)-valued functions

In this section, we use the notations introduced in Definition 5.12.1 and the simple well-known facts listed in the subsequent Proposition 5.12.2, and ℵ will stand for one of the symbols ∞ or ω. We first prove the following left-sided version of the Weierstrass theorem for the group G ℵ (E): 5.14.1 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function Aw : Uw \ {w} → G ℵ (E). Then there exist a holomorphic function H : D \ Z → G ℵ (E) and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, such that Hw Aw = H

on Uw \ {w} .

(5.14.1)

Proof. Let a family of positive integers mw , w ∈ Z, be given. Since G ℵ (E) ⊆ GFIℵ (E), the functions Aw can be interpreted as functions with values in GFIℵ (E). Since the latter group is the group of invertible elements of a Banach algebra and since this group is connected, we can apply Theorem 5.13.1 to it and obtain a holomorphic function H : D \ Z → GFIℵ (E) and holomorphic functions Hw : Uw → GFIℵ (E), w ∈ Z, such that Hw Aw = H

on Uw \ {w}.

(5.14.2)

If dim E < ∞ and therefore G ℵ (E) = GL(E) = GFIℵ (E), this completes the proof. Let dim E = ∞, and let λw : Uw → C, λ : D \ Z → C, Kw : Uw → F ℵ and K : D \ Z → F ℵ be the holomorphic functions with Hw = λw I + Kw and H = λI + K. Then, passing to the factor algebra FI (E)/F ℵ (E) ∼ = C, we see: Since H and Hw are invertible, the functions λ and λw have no zeros, and from (5.14.2) it follows that λw = λ on Uw . Hence the Hw /λ and H/λ are G ℵ -valued functions with the required properties. Precisely in the same way, solely replacing the left-sided Theorem 5.13.1 by the right-sided Theorem 5.13.3, we get the corresponding right-sided result:

5.15. Comments

157

5.14.2 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function Aw : Uw \ {w} → G ℵ (E). Then there exist a holomorphic function H : D \ Z → G ℵ (E) and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, such that Aw H w = H

on Uw \ {w} .

(5.14.3)

Both theorems together again give a two-sided version: 5.14.3 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and two holomorphic functions Fw , Gw : Uw \ {w} → G ℵ (E). Then there exist a holomorphic function H : D \ Z → G ℵ (E) and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, such that Fw Hw Gw = H

on Uw \ {w} .

(5.14.4)

Proof. Let a family of positive integers mw , w ∈ Z, be given. Then from the leftsided Theorem 5.14.1 we get a holomorphic function H l : D \ Z → G ℵ (E) and holomorphic functions Hwl : Uw → G ℵ , w ∈ Z, such that Hwl Gw = H l

on Uw \ {w}.

From the right-sided Theorem 5.14.2 we get a holomorphic function H r : D \ Z → G ℵ (E) and holomorphic functions Hwr : Uw → G ℵ (E), w ∈ Z, such that Fw Hwr = H r

on Uw \ {w}.

Setting H = H l H r and Hw = Hwr Hwl , w ∈ Z, we get holomorphic functions H : D \ Z → G ℵ (E)) and Hw : Uw → G ℵ (E), w ∈ Z, such that Fw Hw Gw = Fw Hwr Hwl Gw = H r H l = H

5.15

on Uw \ {w}.

Comments

This chapter is based on the theory of of multiplicative cocycles (fiber bundles) on Stein manifolds (any domain in C is a Stein manifold), which was developed in the 1950s by H. Grauert [Gr1, Gr2, Gr3] for cocycles with values in a (finite dimensional) complex Lie group and generalized by L. Bungart [Bu] to cocycles with values in a Banach Lie group. The Oka-Grauert principle was first discovered by K. Oka [Ok] for cocycles of several complex variables with values in C∗ . That the Oka-Grauert principle holds also for non-commutative groups is a very deep result due to H. Grauert [Gr1, Gr2, Gr3] (not easy even for one variable). In the papers [GL1, GL2, GL3], the authors presented some of these results, which are relevant for operator theory. Direct proofs for the case of one variable

158


and also some new results were given there. This chapter is an extension and completion of the material contained in these papers. Here style and presentation are essentially improved. The first versions of the Weierstrass theorems from section 3.8 were published in the papers of I. Gohberg and L. Rodman in [GR2] (for matrices) and [GR2] (for infinite dimensional operators).

Chapter 6

Families of subspaces Let E be a Banach space. We denote by G(E) the set of closed subspaces of E. In this chapter we study functions with values in G(E). Consider first the case E = Cn , n ∈ N∗ . Let G(k, n) be the set of kdimensional subspaces of Cn , 0 ≤ k ≤ n. Then G(Cn ) is the disjoint union of the sets G(k, n), and it is well known (see, e.g., [HaGr]) that each of them is a complex manifold called the (complex) Grassmann manifold of k-dimensional subspaces of Cn . From this general point of view it is therefore clear what a holomorphic function with values in G(Cn ) is. For a more direct definition (not using Grassmann manifolds) of holomorphic G(Cn )-valued functions, we refer to section 18.1 of the book [GLR]. Note that all values of such a holomorphic function have the same dimension, except for the case that the domain of definition is not connected. If dim E = ∞, then there are different reasonable definitions of holomorphic G(E)-valued functions. We will discuss them in this chapter. First we introduce the notion of continuous G(E)-valued functions. For that we use the gap metric on G(E).

6.1

The gap metric

Let E be a Banach space. Recall that for two non-empty subsets X, Y ⊂ E, the number dist(X, Y ) = inf x − y (6.1.1) x∈X,y∈Y

is called the distance between X and Y (here · is the norm of E). Note that the distance between two subspaces of E is always zero, because any subspace contains the zero-vector. For subspaces, there is another “distance” which is called the gap (in order to avoid confusion with the distance):

160

Chapter 6. Families of subspaces

6.1.1 Definition. Let E be a Banach space. If X is a subspace of E (not necessarily closed), then we denote by S(X) := x ∈ X x = 1 the unit sphere of X. If X, Y are two subspaces, then we define the gap Θ(X, Y ) between X and Y as follows: If X = {0} and Y = {0}, then θ(X, Y ) := max sup dist v, S(Y ) , sup dist v, S(X) . (6.1.2) v∈S(X)

v∈S(Y )

If X = {0} and Y = {0} or if X = {0} and Y = {0}, then we set θ(X, Y ) = 1, and if X = Y = {0}, then we set θ(X, Y ) = 0. From this definition it is immediately clear that θ(X, Y ) ≤ 2 for all X, Y ∈ G(E). (6.1.3) If X is a subspace of a Banach space E and X is the closure of X, then it is clear that Θ(X, X) = 0. Therefore, on the set of all subspaces of E, the gap is not a metric. However we have: 6.1.2 Theorem. Let E be a Banach space, and G(E) the set of all closed subspaces of E. Then the gap Θ defined by (6.1.1) is a complete metric on G(E). Proof. First we check the three axioms of a metric. I. Suppose X, Y ∈ G(E) and θ(X, Y ) = 0. If at least one of the spaces X and Y is the zero space, then this clearly implies (by the definition of θ(X, Y ) in this case) that also the other one is the zero space and hence X = Y . Now let X = {0} and Y = {0}. Then, in particular, sup dist x, S(Y ) = 0, x∈S(X)

i.e., dist x, S(Y ) = 0 for all x ∈ S(X). Since S(Y ) is closed, this is possible, if and only if, S(X) ⊆ S(Y ). In the same way we see that S(Y ) ⊆ S(X). Hence S(X) = S(Y ), which means that X = Y . II. If we interchange the letters X and Y in the expression of the righthand side of definition (6.1.2), then the value of this expression does not change. Therefore it is clear that θ(X, Y ) = θ(Y, X) for all X, Y ∈ G(E) with X = {0} and Y = {0}. III. We prove the triangle inequality. Let X, Y, Z ∈ G(E). First let Z = {0}. If X = {0} and Y = {0}, then θ(X, Z) = 1 and θ(Z, Y ) = 1 (by definition). Since θ(X, Y ) ≤ 2 (see (6.1.3)), this implies that θ(X, Y ) ≤ 2 = θ(X, Z) + θ(Z, Y ). If X = {0} and Y = {0}, then θ(X, Y ) = 1, θ(X, Z) = 0 and θ(Z, Y ) = 1 which implies that θ(X, Y ) = 1 = 0 + 1 = θ(X, Z) + θ(Z, Y ).

6.1. The gap metric

161

Correspondingly we proceed if X = {0} and Y = {0}. Now let Z = {0} and at least one of the spaces X, Y is the zero space. If X = {0} and Y = {0}, then θ(X, Y ) = 1 and θ(X, Z) = 1 and therefore θ(X, Y ) = θ(X, Z) ≤ θ(X, Z) + θ(Z, Y ). Correspondingly we proceed if X = {0} and Y = {0}. If X = Y = {0}, then θ(X, Y ) = 0 and therefore θ(X, Y ) = 0 ≤ θ(X, Z) + θ(Z, Y ). Finally we consider the case when none of the spaces X, Y, Z is the zero space. Then at least one of the following relations is true: θ(X, Y ) = sup dist v, S(Y ) or θ(X, Y ) = sup dist v, S(X) . v∈S(Y )

v∈S(X)

We may assume that this is the first one (otherwise we have to change the roles of X and Y ). Now let ε > 0. Then we can choose x ∈ S(X) such that θ(X, Y ) ≤ dist x, S(Y ) + ε. (6.1.4) Moreover, then we can take z ∈ S(Z) with dist x, S(Z) + ε ≥ x − z .

(6.1.5)

Then, by the triangle inequality in E, dist x, S(Y ) =

inf v∈S(Y )

x − v ≤

inf

v∈S(Y )

≤ x − z + inf

v∈S(Y )

x − z + z − v

z − v = x − z + dist z, S(Y ) .

By (6.1.4), this implies θ(X, Y ) ≤ x − z + dist z, S(Y ) + ε, and, by (6.1.5), we further get θ(X, Y ) ≤ dist x, S(Z) + dist z, S(Y ) + 2ε ≤ sup dist v, S(Z) + sup dist v, S(Y ) + 2ε ≤ θ(X, Z) + θ(Z, Y ) + 2ε. v∈S(X)

v∈S(Z)

Since ε > 0 can be chosen arbitrarily small, this completes the proof of the triangle inequality for θ. Hence it is proved that θ is a metric. We prove the completeness. Let (Xn )n∈N be a Cauchy sequence in (G(E), θ). Passing to a subsequence we may assume that θ(Xn , Xn+m )
1 − ε. Hence

sup dist y, S(X) ≥ sup dist(y, X) ≥ 1, y∈S(Y )

y∈S(Y )

which implies (6.1.10) by definition of Θ(X, Y ).

6.1.4 Proposition. Let E be a Banach space, and let X, Y be two closed subspaces of E, X = {0}, Y = {0}. Then sup dist(v, Y ) ≤ sup dist v, S(Y ) ≤ 2 sup dist(v, Y ) (6.1.11) v∈S(X)

v∈S(X)

v∈S(X)

164


and

max

sup dist(v, Y ) ,

sup dist(v, X) v∈S(Y )

v∈S(X)

≤ θ(X, Y ) ≤ 2 max

sup dist(v, Y ) ,

sup dist(v, X) . (6.1.12) v∈S(Y )

v∈S(X)

Further, if there exist bounded linear projectors PX and PY from E to X and Y , respectively, then max

sup dist(v, Y ) ,

sup dist(v, X)

≤ PX − PY

(6.1.13)

v∈S(Y )

v∈S(X)

and therefore, by the right inequality in (6.1.12), θ(X, Y ) ≤ 2 PX − PY .

(6.1.14)

Finally, if E is a Hilbert space and PX , PY are the orthogonal projectors from E to X and Y , respectively, then

PX − PY = max

sup dist(v, Y ) , v∈S(X)

sup dist(v, X) .

(6.1.15)

v∈S(Y )

and hence, by (6.1.12),

PX − PY ≤ θ(X, Y ) ≤ 2 PX − PY . Proof. Let ε > 0 be given. Then we can find x ∈ S(X) such that sup dist v, S(Y ) < dist x, S(Y ) + ε.

(6.1.16)

(6.1.17)

v∈S(X)

Further, then we can find y ∈ Y such that

x − y < dist(x, Y ) + ε. Since x = 1, then also

and therefore

(6.1.18)

1 − y < dist(x, Y ) + ε

y y y − y = y 1 − y ≤ dist(x, Y ) + ε.

Since y/ y ∈ S(Y ) and therefore

y , x − dist x, S(Y ) ≤

y

(6.1.19)

6.1. The gap metric

165

now it follows from (6.1.17) that y + ε ≤ x − y + y − y + ε. x − sup dist v, S(Y ) <

y

y v∈S(X) Together with (6.1.18) and (6.1.19) this yields sup dist v, S(Y ) < 2 dist(x, Y ) + 3ε ≤ 2 sup dist(v, Y ) + 3ε. v∈S(X)

v∈S(X)

Since ε > 0 can be chosen arbitrarily small, this proves the right inequality in (6.1.11). The left inequality in (6.1.11) is trivial. (6.1.12) follows from (6.1.11) and the inequality obtained from (6.1.11) interchanging X and Y . Now we assume that there exist bounded linear projectors PX and PY from E to X and Y , respectively. Let ε > 0. Then we can choose x ∈ S(X) such that sup ≤ dist(x, Y ) + ε. v∈S(X)

Since dist(x, Y ) ≤ x − PY x , PX x = x and x = 1, this yields sup dist(v, Y ) ≤ x − PY x = (PX − PY )x + ε ≤ PX − PY + ε. v∈S(X)

Since ε > 0 can be chosen arbitrarily small, this means that sup dist(v, Y ) ≤ PX − PY . v∈S(X)

In the same way we get sup dist(v, X) ≤ PX − PY . v∈S(Y )

Together this implies (6.1.13). Finally we assume that E is a Hilbert space and PX , PY are the orthogonal projectors from E to X and Y , respectively. In view of the general inequality (6.1.13), we only have to prove that

PX − PY ≤ max

sup dist(v, Y ) , v∈S(X)

sup dist(v, X) . v∈S(Y )

First note that, for all v ∈ E with PY v = 0,

(I − PX )PY v = dist(PY v, X) = PY v dist

PY v ,X .

PY v

(6.1.20)

166


Hence

(I − PX )PY v ≤ PY v sup dist (w, X)

for all v ∈ E.

(6.1.21)

for all v ∈ E.

(6.1.22)

w∈S(Y )

In the same way we get

(I − PY )PX v ≤ PX v sup dist (w, Y ) w∈S(X)

Let · , · be the scalar product in E. Since the projectors PX and PY are orthogonal, then

PY (I − PX )v 2 = PY (I − PX )v, PY (I − PX )v = (I − PX )v, PY (I − PX )v

= (I − PX )v, (I − PX )PY (I − PX )v ≤ (I − PX )v

(I − PX )PY (I − PX )v for all v ∈ E. Together with (6.1.21) this yields

PY (I − PX )v 2 ≤ (I − PX )v

PY (I − PX )v sup dist (w, X) w∈S(Y )

and therefore

PY (I − PX )v ≤ (I − PX )v sup dist (w, X)

for all v ∈ E.

(6.1.23)

w∈S(Y )

Since PY − PX = PY (I − PX ) − (I − PY )PX and PY is orthogonal, we get

(PY − PX )v 2 = PY (I − PX )v 2 + (I − PY )PX v 2

for all v ∈ E.

In view of (6.1.23) and (6.1.22) this implies

(PY − PX )v 2 ≤ (I − PX )v

2

2 sup dist (w, X)

w∈S(Y )

≤ v

2

max

+ PX v

2

2 sup dist (w, Y )

w∈S(X)

2

sup dist(w, X), sup dist(w, Y ) w∈S(Y )

for all v ∈ E. Taking the square root we get (6.1.20)

w∈S(X)

6.1.5 Proposition. The set of complemented subspaces1 of a Banach space is open with respect to the gap metric. More precisely: Let E be a Banach space, and let 1A

subspace X of a Banach space E is called complemented if there exists a closed subspace Y of E such that E is the direct sum of X and Y .

6.1. The gap metric

167

X, Y be two closed subspaces of E such that E is the direct sum of X and Y . Let P be the linear projector from E to X parallel to Y , and let Q = I − P . Set ε=

1 . 8 P

Then, for all closed subspaces X , Y of E with θ(X, X ) < ε

and

θ(Y, Y ) < ε,

(6.1.24)

E is the direct sum of X and Y . Moreover, if P is the projector from E to X parallel to Y , then

P ≤ 4 P . (6.1.25) Proof. We may assume that X = {0} and Y = {0}, because if, for example X = {0}, then the inequality θ(X , X) < 1 means, by definition, that X = {0}. Now let X , Y ∈ G(E) with (6.1.24) be given. First prove that X ∩Y = {0}. Assume the contrary. Then we can find v ∈ X ∩Y with v = 1 and, by definition of θ, dist v, S(X) ≤ sup dist w, S(X) ≤ θ(X , X) < ε. w∈S(X )

Therefore we can find x ∈ S(X) with

v − x < ε.

(6.1.26)

In the same way, we find y ∈ S(Y ) with

v − y < ε. Together with (6.1.26) this gives

y − x < 2ε.

(6.1.27)

Since x = 1, this implies

x + y = 2x + (y − x) ≥ 2x − y − x > 2 − 2ε and further, as ε = 1/8 P ≤ 1/8, 1

x + y > 2 − . 4 On the other hand

x + y = P x + Qy = P (x − y) + y ≤ P

x − y + 1, which implies, by (6.1.27),

x + y < P 2ε + 1.

(6.1.28)

168


Since ε = 1/8 P , this means that

x + y
4 P .

and

y =

(6.1.30)

v − P v .

P v

x − x < dist x , S(X) + ε.

Then it follows from the definition of θ that

x − x ≤ sup dist w, S(X) + ε ≤ θ(X , X) + ε.

(6.1.31)

w∈S(X )

Since, by hypothesis θ(X , X ) < ε, this yields

x − x < 2ε.

(6.1.32)

v − P v

v + P v 1 1 ≤ =1+

1 . 2

(6.1.37)

So let a finite dimensional subspace X of E be given, and let ε > 0. Then S(X) is compact. Therefore we can find a finite number of vectors x1 , . . . , xN ∈ S(X) with min x − xj < ε for all x ∈ S(X). (6.1.38) 1≤j≤N

Further, by definition of Θ(X, Y ), we can find y1 , . . . , yN ∈ S(Y ) with yj − xj < Θ(X, Y ) + ε for 1 ≤ j ≤ N.

(6.1.39)

Let F be the span of y1 , . . . , yN . Since Y is of infinite dimension, this is a proper subspace of Y . Therefore we can find y0 ∈ S(Y ) with

dist y0 , F > 1 − ε.

6.1. The gap metric In particular,

171 y0 − yj > 1 − ε

for 1 ≤ j ≤ N. (6.1.40) Since, again by definition of Θ(X, Y ), dist y0 , S(X) ≤ Θ(X, Y ), we can find x0 ∈ S(X) with y0 − x0 < dist y0 , s(X) + ε ≤ Θ(X, Y ) + ε. (6.1.41) Further, by (6.1.38), we can find an index 1 ≤ j0 ≤ N such that x0 − xj0 < ε.

(6.1.42)

From (6.1.39)–(6.1.42) now it follows that Θ(X, Y ) + ε ≥ y0 − x0 ≥ y0 − yj0 − yj0 − xj0 − xj0 − x0 > 1 − ε − Θ(X, Y ) − ε − ε = 1 − 3ε − Θ(X, Y ), i.e., 1 − 2ε. 2 Since ε > 0 is arbitrary, this proves (6.1.37). Θ(X, Y ) >

(ii) Let k ∈ N and let X be a k-dimensional subspace of E. Since X is of finite dimension, there is a continuous linear projector P from E onto X. Then k is the codimension of Ker P in E. Now let X be an arbitrary closed subspace of E with 1 Θ(X, X ) < . 8 P Then it follows from Proposition 6.1.5 (with Y = Y = Ker P ) that E is the direct sum of X and Ker P . Therefore dim X is the codimension of Ker P in E, i.e., dim X = k. (iii) Let k ∈ N and let X be a closed subspace of E which is of codimension k in E. Since X is closed and of finite codimension in E, there is a continuous linear projector P from E onto X. Then dim Ker P = k. Now let X be an arbitrary closed subspace of E with 1 Θ(X, X ) < . 8 P Then it follows from Proposition 6.1.5 (with Y = Y = Ker P ) that E is the direct sum of X and Ker P . Therefore, the codimension of X in E is equal to dim Ker P = k. 6.1.7 Proposition. Let E be a Banach space, let X, Y be closed subspaces of E, and let v ∈ E. Then (6.1.43) dist(v, Y ) − dist(v, X) ≤ 2Θ(X, Y ) v .

172


Proof. It is sufficient to prove that dist(v, Y ) ≤ dist(v, X) + 2Θ(X, Y ) v

(6.1.44)

and then to change the roles of X and Y . Let ε > 0 and x ∈ X with

Since dist(v, X) ≤ v , then

v − x < dist(v, X) + ε.

(6.1.45)

x ≤ 2 v + ε.

(6.1.46)

Further, by definition of Θ(X, Y ), there exists y ∈ Y such that

x − y < Θ(X, Y ) x + ε, which yields, by (6.1.46),

x − y < Θ(X, Y )(2 v + ε) + ε. Together with (6.1.45) this implies that dist(v, Y ) ≤ v − y ≤ dist(v, X) + ε + Θ(X, Y )(2 v + ε) + ε. As ε can be chosen arbitrarily small, this proves (6.1.44).

6.1.8 Proposition. Let E be a Banach space, let (Xn )n∈N be a sequence of subspaces of E (possibly not closed), and let (xn )n∈N be a sequence of vectors xn ∈ Xn which converges to some vector y ∈ E. If there exists a closed subspace Y of E such that lim Θ(Xn , Y ) = 0,

n→∞

then y ∈ Y . Proof. We may assume that xn ∈ S(Xn ) for all n ∈ N. Then, by definition of Θ, we can find a sequence yn ∈ S(Y ), n ∈ N, such that

xn − yn ≤ 2Θ(Xn , Y ) ,

n ∈ N.

Since lim Θ(Xn , Y ) = 0 and lim xn − y = 0, then it follows that also the n→∞

sequence (yn ) converges to Y . As all yn belong to Y and Y is closed, this implies that y ∈ Y .

6.2

Kernel and image of operator functions

6.2.1 Definition. Let E be a Banach space, let D ⊆ C (possibly not open), and let M = {M (z)}z∈D be a family of subspaces of E. M will be called continuous if, for each z ∈ D, the space M (z) is closed in E, and if the map D z → M (z) is

6.2. Kernel and image of operator functions

173

continuous with respect to the gap matric (cf. Theorem 6.1.2). By a section of M over D we mean a vector function f : D → E such that f (z) ∈ M (z) for all z ∈ D. Such a section will be called continuous if it is continuous as a vector function with values in E. Now let E, F be two Banach spaces, let D ⊆ C, and let A : D → L(E, F ) be an operator function. Then we set Im A =

Im A(z)

and

z∈D

Ker A =

Ker A(z)

z∈D

.

The family Im A will be called the image of A, and Ker A will be called the kernel of A. 6.2.2. Let E, F be two Banach spaces, let D ⊆ C, and let A : D → L(E, F ) be an operator function. Suppose A is continuous and, moreover, Im A(z) is closed for all z ∈ D. Then nevertheless it is possible that the families Im A and Ker A are not continuous. For example, assume that D ⊆ C is connected and A : D → L(Cn , Cm ) is a continuous matrix function, n, m ∈ N. If the rank of A(z) is not the same for all z ∈ D, i.e., if the functions D z −→ dim Im A(z)

and

D z −→ dim Ker A(z)

are not constant, then neither Im A nor Ker A is continuous. This follows from Proposition 6.1.6. Hence the constancy of the rank of A is a necessary condition for the continuity of Im A and Ker A. This condition is also sufficient. We prove this in the more general setting of Theorem 6.2.8 below. 6.2.3 Definition. Let E, F be two Banach spaces, and let A ∈ L(E, F ). Then we define kA = 0 if A = 0 and kA =

inf

Av

v∈E,dist(v,Ker A)=1

if A = 0.

(6.2.1)

Recall the following fact which follows easily from the Banach open mapping theorem: kA > 0 if and only if A = 0 and Im A is closed. If this is the case, then the operator A0 : E/ Ker A → Im A induced by A is an invertible operator between the factor space E/ Ker A and Im A and 1 kA = −1 . A 0

(6.2.2)

If E, F are two Banach spaces, D ⊆ C and A : D → L(E, F ) is an operator function, then we denote by kA the function defined by kA (z) = kA(z) , z ∈ D.

174


6.2.4 Lemma. Let E, F be Banach spaces, let A, B ∈ L(E, F ), and let kB > 0. Then

A − B dist v, S(Ker B) ≤ 2 , kB v∈S(Ker A)

A − B sup dist v, S(Im A) ≤ 2 , kB v∈S(Im B) 1 kA ≥ 1 − 2 + 1 A − B − 4Θ Im A, Im B kB , kB

kA ≥ 1 − 2Θ(Ker A, Ker B) kB − A − B . sup

(6.2.3) (6.2.4) (6.2.5) (6.2.6)

Proof of (6.2.3). Assume that

A − B dist v, S(Ker B) > 2 . kB v∈S(Ker A) sup

Then it follows from (6.1.11) in proposition (6.1.4) that A − B dist v, Ker B > . kB v∈S(Ker A) sup

Hence, we can find v ∈ S(Ker A) with dist(v, Ker B) >

A − B . kB

(6.2.7)

Since v = 1 and Av = 0, then

Bv = Bv − Av ≤ B − A . On the other hand, (6.2.7) implies, by definition of kB , that Bv > A − B . Proof of (6.2.4). Assume that

A − B dist v, S(Im A) > 2 . kB v∈S(Im B) sup

Then, it follows from (6.1.11) in proposition (6.1.4) that A − B dist v, Im A > . kB v∈S(Im B) sup

Hence, we can find v ∈ S(Im B) with dist(v, Im A) >

A − B . kB

(6.2.8)


175

Let ε > 0. Since v = 1 and v ∈ Im B, then, by definition of kB , we can find w ∈ E with 1 Bw = v and

w ≤ + ε. kB Then

v − Aw = (B − A)w ≤ B − A

w ≤

B − A + ε B − A . kB

As ε can be chosen arbitrarily small, this implies that

v − Aw ≤

B − A . kB

Hence dist(v, Im A) ≤

B − A , kB

which contradicts (6.2.8). Proof of (6.2.5). Set q=2

1 + 1 A − B + 4Θ(Im A, Im B). kB

We may assume that q < 1, because otherwise (6.2.5) is trivial. We have to prove that then kA > 0 and 1 1 1 ≤ . · kA 1 − q kB By definition (6.2.1) of kA , for that it is sufficient to find, for all ε > 0 and y ∈ Im A, a vector x ∈ E such that 1 1 Ax = y and

x ≤ +ε ·

y . kB 1−q For the latter it is sufficient to construct, for all y ∈ Im A and ε > 0, a sequence (xn )n∈N in E such that, for all n ∈ N, x0 = 0, (6.2.9)

xn ≤ (1/kB + ε) q n−1 y if n ≥ 1 and

n Axj ≤ q n y . y − j=0

Indeed, as 0 ≤ q < 1, then x :=

∞ n=0

xn ∈ E

(6.2.10)

176


exists and we have

x ≤

∞

xn ≤

n=0

and

∞ 1 1 1 n−1 + ε (1 + q) y q = +ε

y kB kB 1−q n=1

n

y − Ax = lim y − Axj ≤ lim q n y = 0, n→∞ n→∞ j=0

i.e., Ax = y. So let y ∈ Im A and ε > 0 be given. Proceeding by induction, we set x0 = 0. It is clear that then (6.2.9) and (6.2.10) are valid for n = 0. Now we assume that, for some k ∈ N, we already have x0 , . . . , xk such that (6.2.9) and (6.2.10) are valid for 0 ≤ n ≤ k. k Then y − j=0 Axj ∈ Im A. Therefore, by the definitions of Θ(Im A, Im B) and kB , we can find xk+1 ∈ E such that k 1 + ε y − Axj

xk+1 ≤ kB j=0 and

k k Axj − Bxk+1 ≤ 2Θ(Im A, Im B)y − Axj . y − j=0

j=0

Since, by induction hypothesis, (6.2.10) holds for n = k, the first inequality proves (6.2.9) for n = k + 1, and the second inequality implies that k Axj − Bxk+1 ≤ 2Θ(Im A, Im B)q k y . y − j=0

As, by definition of q, 2Θ(Im A, Im B) ≤ q/2, this further implies that k 1 Axj − Bxk+1 ≤ q k+1 y . y − 2 j=0

(6.2.11)

Since (6.2.9) is already proved for n = k + 1, we get Axk+1 − Bxk+1 ≤ A − B xk+1 ≤ A − B As, again by definition of q,

A − B

1 +ε kB

≤

q 2

1 + ε q k y . kB


177

(we may assume that ε ≤ 1), this implies that Axk+1 − Bxk+1 ≤ 1 q k+1 y . 2

Together with (6.2.11) this proves (6.2.10) for n = k + 1.

Proof of (6.2.6). Note that Ker B = E, because of kB > 0. We may assume that also Ker A = E, because otherwise Θ(Ker A, Ker B) = Θ(E, Ker B) ≥ 1 (cf. Proposition 6.1.3) and (6.2.6) is trivial. Hence kA is defined by Av . inf kA = v∈E , dist(v,Ker A)=1

Therefore, we have to prove that, for each v ∈ E with dist(v, Ker A) = 1,

Av ≥ 1 − 2Θ(Ker A, Ker B) kB − A − B .

(6.2.12)

(6.2.13)

So let v ∈ E with (6.2.12) be given. Moreover let ε > 0. Then, by (6.2.12), we can find w ∈ Ker A with

v − w < 1 + ε. (6.2.14) As w ∈ Ker A, it follows from (6.2.12) that also dist(v − w, Ker A) = 1.

(6.2.15)

By Proposition 6.1.7, dist(v − w, Ker B) − dist(v − w, Ker A) ≤ 2Θ(Ker A, Ker B) v − w . In view of (6.2.15), this implies that dist(v − w, Ker B) ≥ 1 − 2Θ Ker A, Ker B v − w and further, by definition of kB , B(v − w) ≥ 1 − 2Θ(Ker A, Ker B)v − w kB . Hence

Av = A(v − w) ≥ B(v − w) − A − B

v − w ≥ 1 − 2Θ(Ker A, Ker B)v − w kB − A − B

v − w . By (6.2.14) this implies that

Av ≥ 1 − 2Θ(Ker A, Ker B)(1 + ε) kB − A − B (1 + ε). As ε can be chosen arbitrarily small, this proves (6.2.13).

178


6.2.5 Corollary. Let E, E , F be Banach spaces, and let An ∈ L(E, F ), An ∈ L(E, F ), n ∈ N, be two sequences such that lim An − A0 = lim An − A0 = 0 ,

n→∞

(6.2.16)

n→∞

Ker An = {0} , Im An ⊇

Im An

n ∈ N, n ∈ N,

,

(−1)

and the spaces Im An , n ∈ N, are closed. Let An (−1) operators with An An = I. Then An = A0 lim A(−1) n

(−1)

n→∞

(6.2.17) (6.2.18)

∈ L(Im An , E), n ∈ N, be the

A0 .

(6.2.19)

Proof. Assume the contrary. Then, passing to subsequences we may assume that, for some ε > 0, (−1) An An − A(−1) A0 > ε . 0 Take a sequence of vectors xn ∈ E with xn = 1 and (−1) (An An − A(−1) A0 )xn > ε 0

and xn = 1 ,

n ∈ N∗ .

(6.2.20)

Then

(−1) (−1) An − A0 A0 + An − A0 A0 A0 xn

(An − A0 ) xn = An A(−1) n (−1) (−1) −

A ≥ An A(−1) A − A A − A

A x A n n 0 n n 0 0 . 0 0

Since Ker An = {0}, this implies that (−1) (−1)

(An − A0 ) xn ≥ kAn An(−1) An − A0 A0 xn − An − A0 A0 A0 , and further, by (6.2.20), (−1)

(An − A0 ) xn ≥ kAn ε − An − A0 A0 A0 ,

n ∈ N∗ .

(6.2.21)

Since Ker An = {0}, n ∈ N, it follows from estimate (6.1.45) in Lemma 6.2.4 that kAn ≥ kA0 − An − A0 . Together with (6.2.21) this implies that (−1)

(An − A0 ) xn ≥ kA0 ε − An − A0 ε + A0 A0 , which is a contradiction to (6.2.16).

n ∈ N∗ ,


179

6.2.6 Theorem. Let E be a Banach space and let {M (z}z∈D , D ⊆ C, be a family of subspaces of E such that, for each point z0 ∈ D, there exist a neighborhood U ⊆ D of z0 , Banach spaces X, Y and continuous operator functions T : U → L(X, E), S : U → L(E, Y ) such that, for all z ∈ U , Im S(z) is closed and M (z) = Im T (z) = Ker S(z). Then {M (z}z∈D is continuous. Proof. Let z0 be given, and let U, X, Y, T, S be as in the hypothesis of the theorem. As Im S(z0 ) is closed, and M (z) = Ker S(z) for all z ∈ U , then it follows from (6.2.3) in Lemma 6.2.4 that, for all z ∈ U , S(z) − S(z0 ) dist v, S M (z0 ) ≤ . kS (z0 ) v∈S(M (z)) sup

Since also Im T (z0 ) is closed (as Im T (z0 ) = Ker S(z0 )) and M (z) = Im T (z) for all z ∈ U , from (6.2.4) in Lemma 6.2.4 we get T (z) − T (z0 ) dist v, S M (z) ≤ . kT (z0 ) v∈S(M (z0 )) sup

Together this implies that Θ M (z), M (z0 ) ≤ max

S(z) − S(z0 ) T (z) − T (z0 ) , kS (z0 ) kT (z0 )

.

As both T and S are continuous at z0 , this implies that {M (z}z∈D is continuous at z0 . 6.2.7 Theorem. Let E, F be two Banach spaces, let D ⊆ C, and let A : D → L(E, F ) be a continuous operator function such that, for all z ∈ D, Im A(z) is closed. Then the following three conditions are equivalent: (i) The function kA is continuous. (ii) The family Im A is continuous. (iii) The family Ker A is continuous. (iv) For each compact set K ⊆ D, inf kA (z) > 0.

z∈K

Proof. (i)⇒(iv): Since, by hypothesis of the theorem, Im A(z) is closed for all z ∈ D, kA (z) > 0 for all z ∈ D.

180


Therefore, if kA is continuous, then, for each compact set K ⊆ D, min kA (z) exists z∈K

and is > 0. (iv)⇒(ii) and (iv)⇒(iii): It follows from (6.2.3) in Lemma 6.2.4 that, for all z, w ∈ D, 1 1 , Θ Ker A(z), Ker A(w) ≤ 2 A − B max , kA (z) kA (w) and from (6.2.4) in the same lemma it follows that, for all z, w ∈ D, 1 1 . Θ Im A(z), Im A(w) ≤ 2 A − B max , kA (z) kA (w) If condition (iv) is satisfied, these two inequalities imply that the families Ker A and Im A are continuous. (ii)⇒(i): Set q(z, w) = 2

1 + 1 A(z) − A(w) + 4Θ Im A(z), Im A(w) kA (w)

for z, w ∈ D. Fix z0 ∈ D. Then, by (6.2.5) in Lemma 6.2.4,

z ∈ D, kA (z) ≥ 1 − q(z, z0 ) kA (z0 ) , and

kA (z0 ) ≥ 1 − q(z0 , z) kA (z) ,

z ∈ D.

(6.2.22)

(6.2.23)

Since A is continuous and condition (ii) is satisfied, we have lim q(z, z0 ) = 0.

z→z0

(6.2.24)

By (6.2.22) this implies that, for some ε > 0, kA (z) ≥

kA (z0 ) 2

if z ∈ D and |z − z0 | < ε.

Hence q(z0 , z) ≤ 2q(z, z0 )

if z ∈ D and |z − z0 | < ε,

which implies, by (6.2.24), that also lim q(z0 , z) = 0.

z→z0

(6.2.22)–(6.2.25) together imply that lim kA (z) = kA (z0 ). z→z0

(6.2.25)


181

(iii)⇒(i): Let z0 ∈ D. Then, by (6.2.6) in Lemma 6.2.4, for all z ∈ D, kA (z) ≥ kA (z0 ) − 2Θ Ker A(z), Ker A(z0 ) kA (z0 ) − A(z) − A(z0 ) (6.2.26) and kA (z0 ) ≥ kA (z) − 2Θ Ker A(z), Ker A(z0 ) kA (z) − A(z) − A(z0 ) .

(6.2.27)

Since lim A(z) − A(z0 ) = 0

z→z0

lim Θ Ker A(z), Ker A(z0 ) = 0, (6.2.28)

and

z→z0

it follows from (6.2.26) that, for some ε > 0, kA (z) ≥

kA (z0 ) 2

if z ∈ D and |z − z0 | < ε.

Together with (6.2.27) this implies that kA (z0 ) ≥ kA (z) − 4Θ Ker A(z), Ker A(z0 ) kA (z0 ) − A(z) − A(z0 )

(6.2.29)

if z ∈ D and |z − z0 | < ε. Now, from (6.2.26) and (6.2.28) it follows that lim inf kA (z) ≥ kA (z0 ) z→z0

and from (6.2.29) and (6.2.28) it follows that lim sup kA (z) ≤ kA (z0 ). z→z0

Hence lim kA (z) = kA (z0 ).

z→z0

6.2.8 Theorem. Let E, F be two Banach spaces, let D ⊆ C, and let A : D → L(E, F ) be a continuous operator function such that, for some k ∈ N, at least one of the following conditions is fulfilled: (i) For all z ∈ D, dim Ker A(z) = k and Im A(z) is closed. (ii) For all z ∈ D, the codimension of Im A(z) in F is equal to k. Then both the family Im A and the family Ker A are continuous on D. Proof. First assume that condition (i) is satisfied. Let z0 ∈ D. Since Ker A(z0 ) is of finite dimension, there is a closed subspace X of E such that E is the directsum of Ker A(z0 ) and X. Since also Im A(z0 ) is closed, then the restriction A(z0 )X is a bounded linear isomorphism between X and Im A(z0 ). Hence (6.2.30) c := inf A(z0 )v > 0. v∈S(X)

182


Since A is continuous, we can find an open disc U centered at z0 such that c 2

if z ∈ U ∩ D.

A(z)v > c 2 v∈S(X)

if z ∈ U ∩ D.

Ker A(z) ∩ X = {0}

if z ∈ U ∩ D.

A(z) − A(z0 ) < Then, by (6.2.30), inf

(6.2.31)

In particular, Since dim Ker A(z) = dim Ker A(z0 ) = codim X, this implies that E is the direct sum of Ker A(z) and X if z ∈ U ∩ D. Hence, for z ∈ U ∩ D, A(z)X is a bounded linear isomorphism from X onto Im A. Denote by B(z) the inverse of this isomorphism. Then, by (6.2.31),

B(z)
0 (Def. 6.2.3). Set C=

8 min{kA(z0 ) , 1}

and let U be the open disc with radius r/C centered at z0 . Now let a holomorphic section f : D → Im A be given. Let ∞

f (z) =

fn (z − z0 )n

n=0

be the Taylor expansion of f at z0 , and K := max f (z) . |z−z0 |≤r

Then, by Cauchy’s inequality,

fn ≤

K , rn

n ∈ N.

(6.4.7)

Now it is sufficient to construct a sequence (un )n∈N of vectors un ∈ E such that, for all n ∈ N, n An−k uk = fn (6.4.8) k=0

and

un ≤ K

C n+1 . rn

(6.4.9)

Indeed, then, by (6.4.9), u(z) =

∞

uk (z − z0 )k ,

z ∈ U,

k=0

is a well-defined holomorphic vector function u : U → E, and if A(z)u(z) =

∞

bn (z − z0 )n

n=0

is the Taylor expansion of Au at z0 , then from (6.4.8) it follows that n (Au)(n) (z0 ) 1 n (n−k) A bn = (z0 )u(k) (z0 ) = k n! n! k=0

=

n A(n−k) (z0 ) u(k) (z0 ) k=0

(n − k)!

k!

=

n k=0

An−k uk = fn

for all n ∈ N.

6.4. Holomorphic families of subspaces

189

Hence Au = f on U . To construct the sequence (un )n∈N we proceed by induction. Beginning of induction: Since, by hypothesis, kA(z0 ) > 0 and f0 = f (z0 ) ∈ Im A(z0 ), we can find u0 ∈ E such that A0 u0 = A(z0 )u0 = f0

u0
0. (v) The family Im A is holomorphic. (vi) The family Ker A is holomorphic.

212


Proof. By Theorem 6.2.7, conditions (i)–(iv) are equivalent. Since here the function A is holomorphic, by definition of a holomorphic family of subspaces, (v) is equivalent to (ii) and (vi) is equivalent to (iii). 6.10.2. An operator A ∈ L(E, F ) is called generalized invertible if there exists B ∈ L(F, E) such that ABA = A and BAB = B. The operator B then is called a generalized inverse of A. 6.10.3 Proposition. An operator A ∈ L(E, F ) is generalized invertible, if and only if, there exist projectors P ∈ L(E) and Q ∈ L(F ) such that Im P = Im A and Im Q = Ker A, i.e., if the spaces Im A and Ker A are complemented. Proof. First assume that A has a generalized inverse B. Then (AB)2 = (ABA)B = AB

and

(BA)2 = B(ABA) = BA.

Hence AB and BA are projectors. Obviously, Im(AB) ⊆ Im A, and from A = (AB)A it follows that also Im A ⊆ Im(AB), and thus P := AB is a projector onto Im A. Moreover it is clear that Ker A ⊆ Ker BA and A(BA) = A implies that Ker BA ⊆ Ker A. Thus BA is a projector whose kernel coincides with Ker A. Hence Q := I − BA is a projector onto Ker A. ! := A Conversely, if there exist such projectors P and Q, then the operator −1 ! A Ker Q is an invertible operator from Ker Q onto Im A = Im P . Let A ∈ −1 ! L(Im P, Ker Q) be the inverse of this operator, and set B := A P . Then ABA = !−1 P )A = P A = A. Moreover, then A !−1 P A(I − Q) = I − Q and A !−1 P = (AA −1 ! P , which implies that (I − Q)A

!−1 P A A !−1 P = A !−1 P A (I − Q)A !−1 P BAB = A

!−1 P A(I − Q) A !−1 P = (I − Q)A !−1 P = A !−1 P = B. = A

6.10.4 Theorem. (i) Let A : D → L(E, F ) be holomorphic such that, for all z ∈ D, A(z) is generalized invertible. Suppose the equivalent conditions (i)–(vi) in Proposition 6.10.1 are satisfied. Then there exists a holomorphic function B : D → L(F, E) with ABA = A on D. (ii) Let A : D → L(E, F ) be holomorphic such that all values of A are left invertible. Then there exists a holomorphic function B : D → L(F, E) with BA = I on D. (iii) Let A : D → L(E, F ) be holomorphic such that all values of A are right invertible. Then there exists a holomorphic function B : D → L(F, E) with AB = I on D. Proof. (i) By Proposition 6.10.3, the spaces Im A(z) and Ker A(z) are complemented. Therefore, by Theorem 6.8.5, we have holomorphic functions P : D → L(F ) and Q : D → L(E) such that, for all z ∈ D, the operators P (z) and Q(z)

6.10. One-sided and generalized invertible ...

213

are projectors with Im P (z) = Im A(z) and Ker Q(z) = Ker A(z). Then, for each z ∈ D, A(z) is invertible as an operator from Im Q(z) to Im A(z); let A(−1) (z) be the inverse of this operator. Set B(z) = A(−1) (z)P (z) for z ∈ D. Then it is clear that A(z)B(z)A(z) = A(z) for all z ∈ D. It remains to prove that the function B is holomorphic. Let z0 ∈ D be given. Since Im A and Im Q are Shubin families (Proposition 6.8.4), we can find a neighborhood U of z0 and holomorphic functions T : U → GL(F ) and G : U → GL(E) such that T (z) Im A(z0 ) = Im A(z) Setting

and G(z) Im Q(z0 ) = Im Q(z) ,

S(z) = A(−1) (z0 )T −1 (z)A(z)G(z)

, Im Q(z0 )

z ∈ U.

z ∈ U,

then we get a holomorphic function S : U → GL Im Q(z0 ) . We claim that B(z) = G(z)S −1 (z)A(−1) (z0 )T −1 (z)P (z) ,

z ∈ U.

(6.10.1)

Indeed, since P (z)A(z) = A(z), we get

B(z) − G(z)S −1 (z)A(−1) (z0 )T −1 (z)P (z) A(z)Q(z)

= A(−1) (z)P (z) − G(z)S −1 (z)A(−1) (z0 )T −1 (z)P (z) A(z)Q(z)

= Q(z) − G(z)S −1 (z) A(−1) (z0 )T −1 (z)A(z)G(z) G−1 (z)Q(z) = Q(z) − G(z)S −1 (z)S(z)G−1 (z)Q(z) = 0 ,

z ∈ U.

(6.10.1), as both sides of (6.10.1) vanish on Ker P (z) and This implies Im A(z)Q(z) = Im P (z), B(z)P (z) = S −1 (z)A−1 (z0 )T −1 (z)P (z) ,

z ∈ U.

Since the functions on the right-hand side of (6.10.1) are holomorphic, it follows that B is holomorphic. (ii) Since the values of A are left invertible, Ker A(z) = {0} and Im A(z) is complemented, for all z ∈ D. Hence, by Proposition 6.10.1, Im A is a holomorphic family of complemented subspaces of F , and we can apply part (i) of the theorem. Therefore we have a holomorphic function B : D → L(F, E) with ABA = A on D. Since the values of A are injective, this implies BA = I on D. (iii) Since the values of A are right invertible, Im A(z) = F and Ker A(z) is complemented, for all z ∈ D. Hence, by Proposition 6.10.1, Ker A is a holomorphic family of complemented subspaces of E, and we can apply part (i) of the theorem. Therefore we have a holomorphic function B : D → L(F, E) with ABA = A on D. Since the values of A are surjective, this implies AB = I on D.

214

6.11


Example: A globally non-trivial complemented holomorphic family of subspaces

Here we construct a holomorphic family of complemented subspaces {M (z)}z∈D of a Banach space E which is not “globally continuously injective”, by which we mean that there does not exist a Banach space M0 and a continuous function A : D → L(M0 , E) with Ker A(z) = {0} and Im A(z) = M (z) for all z ∈ D. 6.11.1 Example. We use the notation from example 5.6.2. Additionally here we assume that 1 1 √ . . . > κn are non-zero integers, P1 , . . . , Pn are mutually disjoint finite dimensional projectors1 in E, and P0 = I − P1 − . . . − Pn ; – A+ is a continuous G-valued function on D+ , which is holomorphic in D+ ; – A− is a continuous G-valued function on D− ∪ {∞}, which is holomorphic in D− ∪ {∞} (Section 3.1.1). If Δ ≡ I, then this factorization will be called canonical (see also Definition 7.1.3). By a factorization of A with respect to Γ (without mentioning a group G) we always mean a factorization of A with respect to Γ and GL(E). 7.1.2. The integers κ1 , . . . , κn and the dimensions of the projectors P1 , . . . , Pn in this definition are uniquely determined by A. This will be established in Theorem 7.10.3 below. The numbers κ1 , . . . , κn will be called the non-zero partial indices of A (Def. 7.9.6), and dim Pj will be called the multiplicity of κj (Def. 7.9.8). 7.1.3 Definition. Let G be an open subgroup of the group of invertible elements of a Banach algebra with unit or let G be one of the groups G ∞ (E) or G ω (E), where E is a Banach space (Def. 5.12.1), and let f : Γ → G be continuous. (i) For z0 ∈ Γ, we say that f admits a local factorization at z0 with respect to Γ and G if there exist a neighborhood U of z0 and continuous functions A+ : U ∩ D+ → G and A− : U ∩ D− → G, which are holomorphic in U ∩ D− and U ∩ D+ , respectively, such that A = A− A+

on U ∩ Γ .

(7.1.3)

We say that f admits local factorizations with respect to Γ and G if it admits at each point of Γ a local factorization with respect to Γ and G. 1 By

a projector in E we always mean a bounded linear projector in E. A family P1 , . . . , Pn of projectors in E is called mutually disjoint if Pj Pk = 0 for all 1 ≤ j, k ≤ n with j = k

7.1. Definitions and first remarks about factorization

221

(ii) If g : Γ → G is a second continuous function, then we say that f and g are equivalent with respect to Γ and G if there exist continuous functions h− : D− ∪ {∞} → G and h+ : D+ → G, which are holomorphic in D− ∪ {∞} and D+ , respectively, such that f = h− gh+

on Γ .

(7.1.4)

If, in this case, g ≡ 1, i.e., if (7.1.4) takes the form f = h− h+

on Γ ,

(7.1.5)

then this representation will be called a canonical factorization of f with respect to Γ and G. As we already saw in the scalar case (Remark 3.11.4), not any continuous function admits local factorizations with respect to Γ. However already weak smoothness requirements ensure the existence of local factorizations with respect to Γ. In Section 7.2 we see that Wiener functions admit local factorizations. In Section 7.3 we prove that this is true also for H¨ older functions. 7.1.4 Proposition. Let G an open subgroup of the group of invertible elements of a Banach algebra with unit or let G be one of the groups G ∞ (E) or G ω (E), where E is a Banach space (Def. 5.12.1), and let f : Γ → G be continuous. Assume, at some point z0 ∈ Γ, f admits two local factorizations with respect to Γ and G, i.e., f = h − h+

and

f = h− h+

on Γ ∩ U,

(7.1.6)

h− : U ∩ D− → GA and h+ , h+ : U ∩ where U is a neighborhood of z0 and h− , D+ → G are continuous functions, which are holomorphic in U ∩ D− and U ∩ D+ , respectively. Then there is a holomorphic function g : U → G such that h− = h− g −1

on

D− ∩ U

and

h+ = gh+

on

D+ ∩ U.

(7.1.7)

Proof. By (7.1.6), −1 h−1 − h− = h+ h+

on Γ ∩ U.

Hence, there is a continuous function g : U → G defined by on D− ∩ U, h−1 h− g = − −1 on D+ ∩ U, h+ h + which satisfies (7.1.7) by definition. As this function is holomorphic in U \ Γ, it follows from Theorem 1.5.4 that g is holomorphic on U .

222

Chapter 7. Plemelj-Muschelishvili factorization

7.1.5 Proposition. Let G an open subgroup of the group of invertible elements of a Banach algebra with unit or let G be one of the groups G ∞ (E) or G ω (E), where E is a Banach space (Def. 5.12.1), and let f : Γ → G be continuous. If f admits a canonical factorization f = f+ f− with respect to Γ and G, −1 (∞) and f+ (z) = f+ (z)f− (∞) we get a canonical then setting f− (z) = f− (z)f− factorization f = f− f+ with the additional property f− (∞) = 1. With this additional property, the canonical factorization of f is uniquely determined by f . Proof. Assume there are two canonical factorizations f = f− f+ and f = f− f+ of f with f− (∞) = f− (∞) = 1. Then f− f+ = f− f+ and therefore −1 −1 f− f− = f+ f+

on Γ.

By Theorem 1.5.4, the two sides of this relation define a holomorphic function on C ∪ {∞} with value 1 at infinity. By Liouville’s theorem this implies that f− ≡ f− and f+ ≡ f+ . 7.1.6 Proposition. Let E be a Banach space, and let A : Γ → GL(E) be a continuous function which admits a factorization with respect to Γ and GA. Then there exists always a factorization A = A− ΔA+ with respect to Γ satisfying the additional condition A− (∞) = I. A + is an arbitrary factorization with respect to Γ and GA, − Δ Proof. If A = A then we obtain a factorization A = A− ΔA+ with A− (∞) = I by setting A− (z) = −1 −1 − (∞)A+ (∞). , Δ(z) = A− (∞)Δ(z) A− (∞) and A+ (z) = A A− (z) A− (∞) 7.1.7. Note however that the factorization with the extra condition A− (∞) = I established in Proposition 7.1.6 is not uniquely determined, except for the case of a canonical factorization (Proposition 7.1.5).

7.2

The algebra of Wiener functions and other splitting R-algebras

In this section T is the unit circle, D+ is the open unit disc and D− = C \ D+ . 7.2.1. Let A be a Banach algebra, and let W (A) be the space of Wiener functions with values in A (Def. 3.1.5), i.e., the space of functions f : T → A of the form f (z) =

∞ n=−∞

z n fn

with

f W :=

∞ n=−∞

fn W < ∞.

7.2. The algebra of Wiener functions

223

Then it follows from Cauchy’s product theorem that if f, g ∈ W (A), then the pointwise defined product f g belongs again to W (A), where

f g W ≤ f W g W . Hence W (A) is a Banach algebra. In the present section we prove: If A is a Banach algebra with unit, then each function f ∈ W (A) with invertible values is equivalent (Def. 7.1.3) to some holomorphic function on T. If fact we prove the following stronger result: 7.2.2 Theorem. Let A be a Banach algebra with unit, let GA be the group of invertible elements of A, and let f ∈ W (A) such that f (z) ∈ GA for all z ∈ T. Then: (i) The pointwise defined function f −1 again belongs to W (A). (ii) The function f can be written in the form f = h− hh+ , where: h : T → GA is holomorphic, h− : D− ∪ {∞} → GA is continuous on D− ∪ {∞} and holomorphic in D− ∪ {∞}, h+ : D+ → GA is continuous on D+ and −1 holomorphic in D+ , and the functions h− , h−1 − , h+ and h+ belong to W (A). (The latter statement follows also from (i) and (iii).) (iii) Let g : T → GA be a second function from W (A) which is equivalent to f with respect to Γ and GA (Def. 7.1.3). Further, let g− : D− ∪ {∞} → GA and g+ : D+ → GA be any functions which are holomorphic in D− ∪ {∞} and D+ , respectively, such that g = g− f g+

on T.

(7.2.1)

(Such functions then exist by definition of equivalence.) Then the functions −1 −1 g− , g− , g+ and g+ belong to W (A). The remainder of this section is devoted to the proof of this theorem. In fact we immediately prove a generalization of it (a second interesting example, covered by this generalization, we meet in Section 8.11). 7.2.3 Definition. Let E be a Banach space. By a rational function with values in E or by a rational function f : C → E we mean an E-valued meromorphic function on C (Section 1.10.6) such that f (1/z) is also meromorphic on C. It is easy to see that any rational function with values in E can be written in the form f=

p , ϕ

where p is a holomorphic polynomial with coefficients in E, and ϕ is a scalar holomorphic polynomial. We shall say that ∞ is a pole of an E-valued rational function f if 0 is a pole of the function f (1/z) (Section 1.10.6).

224


7.2.4. Let A be a Banach algebra with the norm · . Then we are interested in Banach algebras R with the norm · R , which are algebraic subalgebras of the algebra of continuous functions f : T → A and which satisfy the following three conditions: (A) max f (z) ≤ f R for all f ∈ R. z∈T

(B) The algebra of all A-valued rational functions without poles on T is contained in R as a dense subset. (C) Each f ∈ R admits a splitting f = f+ + f− with respect to T (Def. 3.1.2), where f− , f+ ∈ R. Sometimes such algebras will be called splitting R-algebras. Obviously, the Wiener algebra W (A) (Section 7.2.1) is a splitting R-algebra. 7.2.5 Theorem. Let A be a Banach algebra with unit, let GA be the group of invertible elements of A, and let R be a Banach algebra of continuous functions f : T → A satisfying conditions (A), (B) and (C) from Section 7.2.4. Let f ∈ R such that f (z) ∈ GA for all z ∈ T. Then: (i) The pointwise defined function f −1 again belongs to R. (ii) The function f can be written in the form f = h− hh+ ,

(7.2.2)

where h− : D− ∪ {∞} → GA is continuous on D− ∪ {∞} and holomorphic in D− ∪ {∞}, h+ : D+ → GA is continuous on D+ and holomorphic in D+ , h is an A-valued rational function without poles on T, h(z) ∈ GA for all z ∈ T, −1 and the functions h− , h−1 − , h+ and h+ belong to R. (The latter statement follows also from (i) and (iii).) (iii) Let g : T → GA be a second function from R which is equivalent to f with respect to Γ and GA (Def. 7.1.3). Further, let g− : D− ∪ {∞} → GA and g+ : D+ → GA be any functions which are holomorphic in D− ∪ {∞} and D+ , respectively, such that g = g− f g+

on T.

(7.2.3)

(Such functions then exist by definition of equivalence.) Then the functions −1 −1 g− , g− , g+ and g+ belong to R. We first prove the corresponding generalization of Proposition 3.10.1: 7.2.6 Proposition. Let R be a Banach algebra of scalar continuous functions f : T → C satisfying conditions (A), (B) and (C) from Section 7.2.4 (with A = C). If f ∈ R and f (z) = 0 for all z ∈ T, then f −1 ∈ R.


225

Proof. Let f ∈ R be given, which is not an invertible element of R. We have to find θ ∈ T with f (θ) = 0. Since f is not invertible, by the theory of commutative Banach algebras, there exits a multiplicative functional Φ on R with Φ(f ) = 0. By condition (B), for all fixed complex numbers λ ∈ C \ T, the function z − λ is an invertible element of R. Hence, for each fixed λ ∈ C \ T, Φ(z − λ) = 0 and, therefore, Φ(z) − λ = Φ(z − λ) = 0. Hence θ := Φ(z) ∈ T. Since Φ is multiplicative and linear, this implies that, for each scalar rational function r without poles on T (by condition (B) such functions belong to R), we have r(θ) = Φ(r). Since the rational functions are dense in R (condition (B)) and by condition (A), this further implies that g(θ) = Φ(g) for all g ∈ R. Hence f (θ) = Φ(f ) = 0. Further, the proof of Theorem 7.2.5 is based on the following 7.2.7 Lemma. Let B be a Banach algebra with unit, and let L be a maximal left ideal of B. Further, let Z be a closed subalgebra of B such that 1 ∈ Z and zb = bz for all b ∈ B and z ∈ Z (i.e., Z is a subalgebra of the center of B with 1 ∈ Z). Then L ∩ Z is a maximal ideal of Z. Proof. Here the unit element of B will be denoted by e. First we prove: Proposition ♣: For each a ∈ Z \ L we have: (i) There exits y ∈ B with ya − e ∈ L. (ii) If y1 , y2 ∈ B with y1 a − e ∈ L and y2 a − e ∈ L, then y1 − y2 ∈ L. Part (i) follows from the hypotheses that L is maximal among the proper left ideals of B. (Otherwise the set of elements of the form l + ya, l ∈ L, y ∈ B, would be a bigger proper left ideal.) For part (ii), we first prove that (7.2.4) L = y ∈ B ya ∈ L . Let Ka be the set on the right-hand side of (7.2.4). It is clear that Ka is a left ideal of B. Since 1 ∈ Ka , Ka is even a proper left ideal of B. If l ∈ L, then la = al ∈ L (as a belongs to the center of B and L is a left ideal). Hence L ⊆ Ka . Since L is maximal among the proper left ideals, this means that L = Ka . Now let y1 , y2 ∈ B with y1 a − e ∈ L and y2 a − e ∈ L be given. Then (y1 − y2 )a = y1 a − e − (y2 a − e) ∈ L, which implies by (7.2.4) that y1 − y2 ∈ L. Proposition ♣ is proved.

226


Now we prove that Z ∩ L is a maximal ideal of Z. It is clear that Z ∩ L is an ideal of Z. Assume it is not maximal. Then there exists a ∈ Z with a − ze ∈ L for all z ∈ C. Therefore, by part (i) of proposition ♣, for each z ∈ C, we can choose an element f (z) ∈ B with f (z)(a − z) − e ∈ L. (7.2.5) To complete the proof, now it is sufficient to show that f (z) ∈ L

for all z ∈ C,

(7.2.6)

because then from (7.2.5) we get the contradiction e ∈ L. Let B/L be the factor space of B with respect to L, and let π : B → B/L be the canonical projector. (7.2.6) then is equivalent to the relation (π ◦ f )(z) = 0

for all z ∈ C.

Therefore, by Liouville’s theorem, it is sufficient to prove the folllowing Proposition ♠: The function π ◦ f is holomorphic on C and (π ◦ f )(∞) = 0. Setting f ∗ (z) =

−1 1 a −e z z

for |z| >

1

a

we define a B-valued holomorphic function f ∗ such that f ∗ (z)(a − ze) − e = 0 ∈ L

for all |z| >

1 .

a

By (7.2.5) and part (ii) of proposition ♣, this implies that f (z) − f ∗ (z) ∈ L for |z| > 1/ a . Hence π ◦f is equal to π ◦f ∗ for |z| > 1/ a . As f ∗ is holomorphic and f ∗ (∞) = 0, this proves that π◦f is holomorphic for |z| > 1/ a and (π◦f )(∞) = 0. It remains to prove that π ◦ f is holomorphic everywhere on C. Let ξ ∈ C be given, and let U be the open disc centered at ξ and with radius 1/ f (ξ) . Then for all z ∈ U , e − (z − ξ)f (ξ) is an invertible element of B, and, setting

−1 f ξ (z) = f (ξ) e − (z − ξ)f (ξ) we get a holomorphic function f ξ : U → B. Note that f ξ (z) =

∞

k+1 (z − ξ)k f (ξ) ,

z ∈ U.

k=0

Therefore

−1 = −e f ξ (z) (z − ξ)e − f (ξ)


227

and, further, −1 f ξ (z)(a − ze) − e = f ξ (z) (a − ze) + (z − ξe) − f (ξ)

−1 −1 = f ξ (z) (a − ξe) − f (ξ) = f ξ (z) f (ξ) f (ξ)(a − ξe) − e for all z ∈ U . Since, by (7.2.5), f (ξ)(a − ξe) − e ∈ L and L is a follows that f ξ (z)(a − ze) − e ∈ L for all z ∈ U . Again by (7.2.5) (ii) of proposition ♣, this implies that f ξ (z) − f (z) ∈ L for all z π ◦ f = π ◦ f ξ on U . As f ξ is holomorphic in U , this further implies holomorphic in U . Proposition ♠ is proved.

left ideal, it and by part ∈ U . Hence that π ◦ f is

7.2.8. Proof of Theorem 7.2.5 (i): Assume f is not an invertible element of the algebra R. Then the set of elements of the form gf , g ∈ R, is a proper left ideal of R. Let L be a maximal left ideal in R with gf g ∈ R ⊆ L. (7.2.7) Further, let RC be the subalgebra of R, which consists of the functions of the form ψe, where ψ is a scalar function and e is the unit element of A. By Lemma 7.2.7, L ∩ RC is a maximal ideal of RC , and it follows from Proposition 7.2.6 (and the theory of commutative Banach algebras) that there exists a point θ ∈ T such that L ∩ RC = ϕ ∈ RC ϕ(θ) = 0 . (7.2.8) Therefore, for all ϕ ∈ RC , the function ϕ(z) − ϕ(θ),

z ∈ T,

belongs to L ∩ RC . Since L is a left ideal, it follows that, for all a ∈ A and ϕ ∈ L ∩ RC , the function aϕ(z) − aϕ(θ),

z ∈ T,

belongs to L. Hence, for all rational functions r : C → A without poles on T (which belong to R by condition (B)), the function r(z) − r(θ),

z ∈ T,

belongs to L. Now we introduce the function g ∈ R defined by −1 g(z) := f (θ) f (z), z ∈ T. Then, by condition (B), there is a sequence rν : T → A of rational functions without poles on T such that lim g − rν R = 0. ν

228


As g(θ) = e (the unit element of A), then it follows from condition (A), that also lim e − rν (θ) = lim g(θ) − rν (θ) = 0. ν

ν

Hence, the sequence of functions rν (z) − rν (θ),

z ∈ T,

converges to the function g(z) − e,

z ∈ T,

with respect to the norm of R. Since L is closed with respect to this norm and each of the functions rν (z) − rν (θ) belongs to L, it follows that the function g(z) − e,

z ∈ T,

belongs to L. As, by definition of L, also g ∈ L, this implies that the constant function with value e, i.e., the unit element of R, belongs to L. As L is a proper ideal, this is a contradiction. 7.2.9. Let A be a Banach algebra with unit, and let R be a Banach algebra of continuous functions f : T → A satisfying conditions (A), (B) and (C) from Section 7.2.4. Then we denote by R+ the subalgebra of all f ∈ R, which admit a continuous extension to D+ , which is holomorphic in D+ , and by R− we denote the subalgebra of all f ∈ R, which admit a continuous extension to D− ∪ {∞}, which is holomorphic in D− ∪ {∞}. Further, we denote by R0− the subalgebra of all f ∈ R− (A) with f (∞) = 0. Since all constant functions with value in A belong to R (condition (B)), it follows from condition (C) that R is the direct sum of R+ and R0− . We denote by P+ the linear projector from R to R+ parallel to R0− , and we set P− = I − P+ . We denote by P± the operator norms of P± with respect to the norm · R of R:

P± := sup

P± f R . (7.2.9) f ∈R,f R =1

For R = W (A) (Section 7.2.1), obviously, P± = 1. Now, as a special case of the factorization Lemma 5.2.3, we immediately obtain: 7.2.10 Lemma. Let A be a Banach algebra with unit, let R be a Banach algebra of continuous functions f : T → A satisfying conditions (A), (B) and (C) from Section 7.2.4, and let C = max P+ , P− , where P+ , P− are the projectors defined in Section 7.2.9. Then each f ∈ R with

f − 1 R
0, the following holds: (*) For each ϕ ∈ OGA (W ) with maxz∈W ϕ(z) − 1 < ε, there exist ϕ+ ∈ OG (W + ) and ϕ− ∈ OGA (W − ∪ {∞}) with ϕ = ϕ− ϕ+ on W . By Theorem 7.4.2 we may assume that f is a GA-valued holomorphic function on C \ {0}. Let ∞ f (z) = z n fn n=−∞

be the Laurent expansion of f at zero. Choose integers K < M such that the function M u(z) := z n fn n=K

is so close to f over W that u(z) ∈ GA for all z ∈ W and max u−1 (z)f (z) − 1 < ε. z∈W

Then, by statement (*), we can find f+ ∈ OG (W + ) and g− ∈ OG W − ∪ {∞} −1 such that u−1 f = g− f+ on W . Hence the function f f+ = ug− has a Laurent expansion of the form M −1 (z) = z n an . (7.6.4) f (z)f+ n=−∞

242


Choose an integer N such that the function v(z) :=

M

z n an

n=N −1 is so close to f f+ over W that v(z) ∈ GA for all z ∈ W and −1 (z)v −1 (z) − 1 < ε. max f (z)f+ z∈W

we can find g+ ∈ OGA (W + ) and f− ∈ Then, again by statement−1(*), OGA W − ∪ {∞} such that f f+ v −1 = f− g+ on W . Set −1 −1 f f+ . h = f−

Then h ∈ OGA (W ), as f is holomorphic and GA-valued on C \ {0}. Moreover, since h = g+ v, the Laurent expansion of h is of the form h(z) =

∞

z n hn .

n=N −1 −1 f f+ , it follows from (7.6.4) that the Laurent On the other hand, as h = f− expansion of h is of the form M

h(z) =

z n hn .

n=−∞

Hence, the Laurent expansion of h is of the form h(z) =

M

z n hn .

n=N

As f = f− hf+ , this completes the proof.

7.7

The finite dimensional case

In this section, L(n, C), n ∈ N∗ , is the algebra of complex n×n-matrices, GL(n, C) is the group of invertible complex n×n-matrices, D+ ⊆ C is a bounded, connected, open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ , and D− = C \ D+ . Here we prove the following theorem: 7.7.1 Theorem. Let A : Γ → GL(n, C) be a continuous function which admits local factorizations with respect to Γ. Then A admits a factorization with respect to Γ.

7.7. The finite dimensional case

243

Proof. By Theorem 7.6.1 (i) we may assume that A is of the form M

A(z) =

Aj ∈ L(n, C) ,

z j Aj ,

(7.7.1)

j=N

where −∞ < N ≤ M < ∞. We need here the following definition: For κ ∈ Z and x ∈ Cn \ {0}, a pair (ϕ− , ϕ+ ) is called a κ-section of x if ϕ+ : D+ → Cn and ϕ− : D− ∪ {∞} → Cn are holomorphic functions such that z κ ϕ− (z) = A(z)ϕ+ (z)

for all z ∈ Γ

and

ϕ+ (0) = x.

If κ ≥ M , then, for each x ∈ Cn \ {0}, there exists a κ-section of x, namely ϕ+ (z) := x

ϕ− (z) := z −κ A(z)x =

and

M −κ

z j Aj+κ x .

j=N −κ

On the other hand, if κ ≤ N − 1, then no x ∈ Cn \ {0} has a κ-section. Indeed, let κ ≤ N − 1, and let (ϕ− , ϕ+ ) be a κ-section of a vector x ∈ Cn \ {0}. Then −

ϕ (z) = z

−κ

M

z Aj ϕ+ (z) j

for all z ∈ Γ .

j=N

Then the left-hand side of this equation is holomorphic on D− ∪ {∞} and, as j − κ ≥ 1 for N ≤ j ≤ M , the right-hand side is holomorphic on D+ . Hence, by Liouville’s theorem, both sides vanish identically. It follows that +

A(z)ϕ (z) =

M

z Aj ϕ+ (z) = 0 j

for z ∈ Γ .

j=N

Since the values of A on Γ are invertible, this implies that ϕ+ = 0 on Γ. Hence, by uniqueness of holomorphic functions, ϕ+ ≡ 0 on D+ . In particular x = ϕ+ (0) = 0. Hence, for each x ∈ Cn \ {0}, there exists a smallest integer κ such that x has a κ-section. We denote this integer by κ(x). Note that, for each κ ∈ Z, the set 0 ∪ x ∈ Cn \ {0} κ(x) ≤ κ is a linear subspace of Cn . Therefore, we can find a basis e1 , . . . , en of Cn such that κ(en ) = κ(ej ) =

min

x∈Cn \{0}

κ(x),

min

x∈Cn \span(ej+1 ,...,en )

κ(x)

for 1 ≤ j ≤ n − 1.

(7.7.2)

244


+ Fix a κ(ej )-section (ϕ− j , ϕj ) of ej for each j. Denote by A+ and A− the matrices + formed by the columns ϕj and ϕ− j , respectively, and let Δ be the diagonal matrix κ(e1 ) κ(en ) ,...,z . Then with diagonal z

ΔA− = AA+ ,

(7.7.3)

and A+ (0) is the matrix with the columns e1 , . . . , en . It remains to prove that A+ (z) is invertible for all z ∈ D+ , and A− (z) is invertible for all z ∈ D− ∪ {∞}. To prove this we assume the contrary. First assume that the matrix A+ (z0 ) is not invertible for some z0 ∈ D+ . Since A+ (0) is invertible, z0 = 0. Then there exist 1 ≤ k ≤ n and numbers λk , . . . , λn with n λ j ϕ+ and λk = 0, j (z0 ) = 0 j=k

and there is a holomorphic function ψ + on D+ with ψ + (z) =

n 1 λ j ϕ+ j (z) z − z0

for z ∈ D+ \ {z0 } .

j=k

Moreover, since κ(e1 ) ≥ . . . ≥ κ(en ), there is a holomorphic function ψ − on D− ∪ {∞} with ψ − (z) =

n z λj z κ(ej )−κ(ek ) ϕ− j (z) z − z0

for z ∈ D− .

j=k

+ Since (ϕ− j , ϕj ) is a κ(ej )-section of ej , we get n n 1 1 + A(z)ψ (z) = λj A(z)ϕj (z) = λj z κ(ej ) ϕ− j (z) z − z0 z − z0 +

j=k

= z κ(ek )−1

z z − z0

n

j=k

κ(ek )−1 − λj z κ(ej )−κ(ek ) ϕ− ψ (z) j (z) = z

for z ∈ Γ.

j=k

Hence (ψ − , ψ + ) is a (κ(ek ) − 1)-section of the vector ψ + (0) = −

n 1 λj ej . z0 j=k

Since λk = 0, this vector belongs to Cn \ span(ek+1 , . . . , en ), which is a contradiction to (7.7.2). Now we assume that the matrix A− (z0 ) is not invertible for some z0 ∈ D− ∪ {∞}. Then there exist 1 ≤ k ≤ n and numbers λk , . . . , λn with n j=k

λ j ϕ− j (z0 ) = 0

and λk = 0,

7.8. Factorization of G ∞ (E)-valued functions

245

and there is a holomorphic function ψ − on D− ∪ {∞} with

ψ − (z) =

⎧ n z ⎪ ⎪ λ j ϕ− ⎨ z−z0 j (z)

if z0 ∈ D− and z ∈ D− \ {z0 } ,

n ⎪ ⎪ λ j ϕ− ⎩z j (z)

if z0 = ∞ and z ∈ D− .

j=k

j=k

Moreover, since κ(e1 ) ≥ . . . ≥ κ(en ), there is a holomorphic function ψ + on D+ with ⎧ n 1 ⎪ ⎪ λj z κ(ek )−κ(ej ) ϕ+ if z0 ∈ D− and z ∈ D+ , ⎨ z−z0 j (z) j=k + ψ (z) = n ⎪ ⎪ λj z κ(ek )−κ(ej ) ϕ+ if z0 = ∞ and z ∈ D+ . ⎩ j (z) j=k

+ Using again that (ϕ− j , ϕj ) is a κ(ej )-section of ej , for all z ∈ Γ we get

+

A(z)ψ (z) =

⎧ n 1 κ(ek )−1 − ⎪ ⎪ λj z κ(ek ) ϕ− ψ (z) ⎨ z−z0 j (z) = z

if z0 ∈ D− ,

n ⎪ κ(ek )−1 − ⎪ λj z κ(ek ) ϕ− ψ (z) ⎩ j (z) = z

if z0 = ∞ .

j=k

j=k

Hence (ψ − , ψ + ) is a (κ(ek )−1)-section of the vector ψ + (0). Let m be the index with k ≤ n ≤ m such that κ(ej ) = κ(ek ) for k ≤ j ≤ m and, if m < n, κ(em+1 ) < κ(ek ). Then ⎧ m ⎪− 1 λ e ⎪ if z0 ∈ D− , j j ⎨ z0 j=k + ψ (0) = m ⎪ ⎪ λj ej if z0 = ∞ . ⎩ j=k

Since λk = 0, in both cases ψ + (0) belongs to Cn \ span(ek+1 , . . . , en ), which is again a contradiction to (7.7.2).

7.8

Factorization of G ∞ (E)-valued functions

In this section E is a Banach space with dim E = ∞, D+ ⊆ C is a bounded, connected, open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ , and D− = C \ D+ . Here we study the factorization problem for functions with values in G ∞ (E) ω or G (E) (Def. 5.12.1). For G ∞ (E) we obtain the complete solution (Theorem 7.8.6) at the end of this section. For G ω (E) this is more difficult and can be done only later (Theorem 8.2.2). But a part of the argument works for both G ∞ (E) and G ω (E). We start with these points of the argument.

246


7.8.1 Proposition. Let ℵ = ∞, ω, and let A : Γ → G ℵ (E) be a continuous function which admits a factorization with respect to Γ. Let A = A− ΔA+ be a factorization of A with respect to Γ and GL(E). Assume moreover that A− (∞) = I (see Proposition 7.1.6). Then, automatically, the values of A− and A+ belong to G ℵ (E). Proof. Denote by 1 the unit element in the factor algebra L(E)/F ℵ (E), and let π : L(E) → L(E)/F ℵ (E) be the canonical map. Then π(A) = π(Δ) = 1 and therefore 1 = π(A− )π(A+ ) on Γ and π A− (∞) = 1 . Hence

π A−1 = π A+ −

and

on Γ

π A−1 − (∞) = 1 .

(7.8.1) (7.8.2)

By Theorem 1.5.4 the two sides of (7.8.1) define a holomorphic function f : C ∪ {∞} → L(E)/F ℵ (E). By (7.8.2), f (∞) = 1. It follows from Liouville’s theorem that f ≡ 1, i.e., π A−1 and π A+ = 1 on D+ . = 1 on D− ∪ {∞} − This means that the values of A− and A+ belong to G ℵ (E). 7.8.2 Theorem. Let ℵ = ∞, ω, and let G L(E)/F ℵ (E) be the group of invertible elements of the factor algebra L(E)/F ℵ (E), and let π : L(E) → L(E)/F ℵ (E) be the canonical map. Further, let W ⊆ C be an open set, and let f : D → G L(E)/F ℵ (E) be holomorphic. Then there exists a holomorphic function A : D → GL(E) such that f (z) = π A(z) for all z ∈ W. (7.8.3) Proof. We first prove this locally. Consider an arbitrary point z0 ∈ W . Let f (z) =

∞

(z − z0 )n fn

n=0

be the Taylor expansion of f at z0 . Since π is bounded and surjective, then, by the Banach open mapping theorem, we can find operators Fn ∈ L(E) with π(Fn ) = fn and Fn < 2 fn . Let U ⊆ W be an open disc around z0 . Then, setting F (z) =

∞ n=0

(z − z0 )n Fn ,

z ∈ U,

we get a holomorphic function F : U → L(E) with π F (z) = f (z) for all z ∈ U . Since f (z0 ) is invertible, then F0 is a Fredholm operator with index zero. Therefore


247

we can find a finite dimensional operator K in E such that F0 + K is invertible. Choose a neighborhood V ⊆ U so small that F (z) + K is invertible for all z ∈ V . Setting A0 (z) = F (z)+ K, z ∈ V , we obtain a holomorphic function A0 : V → GL(E) with f (z) = π A0 (z) for z ∈ V . By the local statement just proved, there exist an open covering U = {Uj }j∈I of D and holomorphic functions Aj : Uj → GL(E) with π Aj (z) = f (z) for z ∈ Uj . Then π Aj A−1 = π f f −1 = 1 on Uj ∩ Uk k for all j, k ∈ I with Uj ∩ Uk = ∅. Therefore, the values of the functions Aj A−1 k lie in G ℵ (E). Now, from Theorem 5.12.5 we get a family of functions Vj : Uj → G ℵ (E) with −1 Aj A−1 on Uj ∩ Uk k = Vj Vk for all j, k ∈ I with Uj ∩ Uk = ∅. Setting A = Vj−1 Aj on Uj we complete the proof. 7.8.3 Lemma. Let ℵ = ∞, ω, and let A : Γ → G ℵ (E) be a continuous function which admits local factorizations with respect to Γ and GL(E) (Def. 7.1.3). Then A admits local factorizations with respect to Γ and G ℵ (E). Proof. Let w ∈ Γ be given. By hypothesis there exist a neighborhood U of w and − : U ∩ D− → GL(E) and A + : U ∩ D+ → GL(E), which continuous functions A are holomorphic in U ∩ D− and U ∩ D+ , respectively, such that + − A A=A

on U ∩ Γ .

(7.8.4)

Let G L(E)/F ℵ (E) be the group of invertible elements of the factor algebra L(E)/F ℵ (E), let 1 be its unit element, and let π : L(E) → L(E)/F ℵ (E) be the canonical map. Since π(A) = 1, then it follows from (7.8.4) that −1 + π A =π A on U ∩ Γ . (7.8.5) − Hence, by Theorem 1.5.4, there is a holomorphic function f : U→ G L(E)/F ∞ (E) with −1 + . f U ∩D = π A = π A and f (7.8.6) − U ∩D − + By Theorem 7.8.2 we can find a holomorphic function A : U → GL(E) such that π(A) = f .

(7.8.7)

+ on U ∩ D+ . Then it follows from − A on U ∩ D− and A+ = A−1 A Set A− = A (7.8.4) that A = A− A+ on Γ, and from (7.8.7) and (7.8.6) it follows that π(A− ) = 1 on U ∩ D− and π(A+ ) = 1 on U ∩ D+ , i.e., the values of A− and A+ belong to G ℵ (E).

248


7.8.4 Proposition. Let ℵ = ∞, ω, let 0 < r < R < ∞ such that D+ ⊆ W := z ∈ C r < |z| < R ,

(7.8.8)

and let A : Γ → G ℵ (E) be a continuous function which admits local factorizations with respect to Γ and GL(E) (Def. 7.1.3). Then: (i) The function A can be written in the form A = A− HA+ ,

(7.8.9)

where the functions A± and H have the following properties: A− : D− ∪ {∞} → G ℵ (E) is continuous on D− ∪ {∞} and holomorphic in D− ∪ {∞}; A+ : D+ → G ℵ (E) is continuous on D+ and holomorphic in D+ ; H is of the form H(z) = I +

M

z n Hn ,

N, M ∈ Z,

Hn ∈ F ℵ (E),

(7.8.10)

n=N

and H(z) ∈ G ℵ (E) for all z ∈ W ; (ii) The function A can be written in the form A = A− HA+ ,

(7.8.11)

where the functions A± and H have the following properties: A− : D− ∪ {∞} → G ℵ (E) is continuous on D− ∪ {∞} and holomorphic in D− ∪ {∞}; A+ : D+ → G ℵ (E) is continuous on D+ and holomorphic in D+ ; H −1 is of the form H −1 (z) = I +

M

z n Hn ,

N, M ∈ Z,

Hn ∈ F ℵ (E),

(7.8.12)

n=N

and H −1 (z) ∈ G ℵ (E) for all z ∈ W . Proof. The proofs of parts (i) and (ii) are similar (for (i) we use part (i) of Theorem 7.6.1, and for (ii) we use part (ii) of that theorem). We restrict ourselves to part (i). Let FIℵ (E) be the Banach algebra of Definition 5.12.1. By Lemma 7.8.3, A admits also local factorizations with respect to Γ and GFIℵ (E). Therefore, we can apply part (i) of Theorem 7.6.4 (with A = FIℵ (E)), and we obtain a representation of A in the form A = a− ha+ , where


249

a− : D− ∪ {∞} → GFIℵ (E) is continuous on D− ∪ {∞} and holomorphic in D− ∪ {∞}; a+ : D+ → GFIℵ (E) is continuous on D+ and holomorphic in D+ ; h is of the form h(z) =

M

z n hn ,

N, M ∈ Z,

n=N

and h(z) ∈ GFIℵ (E) for all z ∈ W . Let V : Γ → F ℵ (E), V− : D− ∪ {∞} → F ℵ (E), V+ : D+ → F ℵ (E), U : W → F ℵ (E), λ− : D− ∪ {∞} → C∗ , λ+ : D+ ∪ {∞} → C∗ and λ : W → C∗ be the functions with A = I + V , a± = λ± I + V± and h = λI + U . Then I + V = λ− I + V− λI + U λ+ I + V+ , which implies (as dim E = ∞) that λ−1 = λ− λ+ . Hence V+ V− −1 I +λ U I + . A= I+ λ− λ+ −1 h, we conclude the proof. Setting A± = λ−1 ± a± and H = λ

The following Lemma 7.8.5 as well as the subsequent theorem here will be proved only for the group G ∞ (E). Below we obtain these results also for G ω (E) (Theorem 8.2.2). But then the proof is more difficult, because then the approximation argument used in the proof of Lemma 7.8.5 does not work. 7.8.5 Lemma. Let A : Γ → G ∞ (E) be a continuous function which admits local factorizations with respect to Γ and GL(E) (Def. 7.1.3). Moreover, let 0 < r < R < ∞ be given such that (7.8.13) D+ ⊆ W := z ∈ C r < |z| < R . Then A is equivalent with respect to Γ and G ∞ (E) to a function of the form Q + P AP P , where: – P is a finite dimensional projector in E and Q = I − P ; – AP is a rational function with values in L(Im P ) such that AP (z) is invertible for all z ∈ W . Proof. By part (i) of Proposition 7.8.4, we may assume that A is already of the form M A(z) = I + z n An , N, M ∈ Z, An ∈ F ∞ (E), n=N

250


and that A(z) is invertible for all z ∈ W . Set and W+ = z ∈ C |z| < R . W− = z ∈ C r < |z| Then, by Theorem 7.5.1, for some ε > 0, the following holds: (*) For each B ∈ OGL(E) (W ) with maxz∈W B(z) − 1 < ε, there exist B+ ∈ OGL(E) (W + ) and B− ∈ OGl(E) (W − ∪ {∞}) with B = B− B+ on W . As An ∈ F ∞ (E), we can chose finite dimensional operators Fn so close to An that, for the function M F (z) := I + z n Fn n=N

we have: F (z) ∈ GL(E)

for all z ∈ W ,

and

max AF −1 (z) − I < ε. z∈W

Then, by (*), AF −1 can be written in the form AF −1 = B− B+

(7.8.14)

with B+ ∈ OGL(E) (W + ) and B− ∈ OGL(E) (W − ∪ {∞}). Choose a finite dimensional projector in E with Im Fn ⊆ Im P and Ker P ⊆ Ker Fn for N ≤ n ≤ M , and set Q = I − P . Then F = P F P + Q on W . Setting AP = P F Im P , we conclude the proof. 7.8.6 Theorem. Let A : Γ → G ∞ (E) be a continuous function which admits local factorizations with respect to Γ and GL(E). Then A admits a global factorization with respect to Γ and G ∞ (E). Proof. This follows immediately from the preceding Lemma 7.8.5 and Theorem 7.7.1. Under certain additional assumptions about the function A in Theorem 7.8.6, we can say correspondingly more about the factors of the factorizations of A. We have: 7.8.7 Corollary. Let 0 < α < 1 and k ∈ N, where, for k ≥ 1, we additionally assume that Γ is of class C k (Def. 3.4.1). Let A : Γ → G ∞ (E) be a function of class Ck+α (Def. 3.4.3). Then: (i) A admits a factorization with respect to Γ and G ∞ (E). (ii) If A = A− ΔA+ is an arbitrary factorization of A with respect to Γ, then automatically, the factors A− and A+ are of class Ck+α on D− and D+ , respectively.

7.9. Filtration with respect to a contour

251

Proof. Part (i) of Theorem 7.3.1 in particular states that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 7.8.6. Part (ii) of the corollary follows from part (ii) of Theorem 7.3.1 7.8.8 Corollary. Let Γ = T be the unit circle, and let R be a Banach algebra of continuous L(E)-valued functions conditions (A), (B) and (C) in Section satisfying 7.2.4. For example, let R = W L(E) be the Wiener algebra (see Section 7.2.1). Let A : Γ → G ∞ (E) be a function which belongs to R. Then: (i) A admits a factorization with respect to Γ and G ∞ (E). (ii) If A = A− ΔA+ is an arbitrary factorization of A with respect to Γ, then automatically, the factors A− and A+ belong to the algebra R. Proof. Part (ii) of Theorem 7.2.5 in particular states that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 7.8.6. Part (ii) of the corollary follows from part (iii) of Theorem 7.2.5.

7.9

The filtration of an operator function with respect to a contour

In this section E is a Banach space, D+ ⊆ C is a bounded connected open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ . We set D− = C \ D+ . 7.9.1 Definition. Let A : Γ → GL(E) be a continuous function, and let κ ∈ Z. (i) A pair ϕ = (ϕ− , ϕ+ ) will be called a (Γ, κ)-section or simply a κ-section of A if ϕ− : D− ∪ {∞} → E and ϕ+ : D+ → E are continuous E-valued functions which are holomorphic in D− ∪ {∞} and D+ , respectively, such that z κ ϕ− (z) = A(z)ϕ+ (z)

for z ∈ Γ.

(7.9.1)

(ii) We denote by M (κ, A) = M (κ, Γ, A) the space of all κ-sections of A. We consider M (κ, A) as a Banach space endowed with the norm defined by

ϕ :=

max

z∈D − ∪{∞}

ϕ− (z) + max ϕ+ (z)

(7.9.2)

z∈D +

for ϕ = (ϕ− , ϕ+ ) ∈ M (κ, A). (iii) We define M− (z, κ, A) = M− (z, κ, Γ, A) = ϕ− (z) (ϕ− , ϕ+ ) ∈ M (κ, A)

for z ∈ D− ∪ {∞}

252

Chapter 7. Plemelj-Muschelishvili factorization and M+ (z, κ, A) = M+ (z, κ, Γ, A) = ϕ+ (z) (ϕ− , ϕ+ ) ∈ M (κ, A)

for z ∈ D+ .

Since 0 ∈ Γ, it follows from (7.9.1) that M− (z, κ, Γ, A) = A(z)M+ (z, κ, Γ, A)

for all z ∈ Γ and κ ∈ Z.

(7.9.3)

7.9.2. Let A : Γ → E be a continuous function, let κ, μ ∈ Z, and let (ϕ− , ϕ− ) be a κ-section of A. Since Γ is the boundary both of D− and D+ , then it follows by uniqueness of holomorphic functions from (7.9.1) that each of the two components ϕ+ and ϕ− is uniquely determined by the other one. 7.9.3. Let A : Γ → E be a continuous function, and let κ, μ ∈ Z with κ ≥ μ. Then M− (z, κ, Γ, A) ⊇ M− (z, μ, Γ, A)

for z ∈ D− ∪ {∞} ,

(7.9.4)

M+ (z, κ, Γ, A) ⊇ M+ (z, μ, Γ, A)

for z ∈ D+ .

(7.9.5)

This follows from two obvious statements: (i) If (ω− , ω+ ) ∈ M (μ, Γ, A) and z ∈ D− ∪ {∞}, then (φ− , φ+ ) := (ω− , ω + ) with ω + (ζ) := ζ κ−μ ω+ (ζ) ,

ζ ∈ D+ ,

is a κ-section of A with φ− (z) = ω− (z). (ii) If (ω− , ω+ ) ∈ M (μ, Γ, A) and z ∈ D+ , then (φ− , φ+ ) := ( ω− , ω+ ) with ω − (ζ) := ζ μ−κ ω− (ζ) ,

ζ ∈ D− ∪ {∞} ,

is a κ-section of A with φ+ (z) = ω+ (z). 7.9.4 Lemma. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ. Then there exist integers κ− ≤ κ+ such that: (i) If κ ∈ Z with κ < κ− , then M (κ, Γ, A) = 0 ,

(7.9.6)

i.e., M− (z, κ, Γ, A) = 0

for all z ∈ D− ∪ {∞} ,

M+ (z, κ, Γ, A) = 0

for all z ∈ D+ .

(7.9.7)

(ii) If κ ∈ Z with κ ≥ κ+ , then M− (z, κ, Γ, A) = E

for all z ∈ D− ∪ {0} ,

M+ (z, κ, Γ, A) = E

for all z ∈ D+ .

(7.9.8)


253

Proof. By Theorem 7.6.1, there exist integers N ≤ M , functions H, T : Γ → GL(E) of the form S(z) =

M

z n Sn

and

H(z) =

n=N

and functions A− , T− ∈ O such that

z n Hn ,

n=N

GL(E)

and

M

D− ∪{∞} (Def. 5.4.5) and A+ , T+ ∈ OGL(E) D+

A = A− HA+

on Γ

(7.9.9)

A = T− S −1 T+

on Γ .

(7.9.10)

Set κ− = N

and

κ+ = max(−N, M ) .

We now first prove part (i). Let κ ≤ κ− − 1 = N − 1 be given, and let (ϕ− , ϕ+ ) ∈ M (κ, A). Then it follows from (7.9.9) that z κ ϕ− (z) = A− (z)

M

z n Hn A+ (z)ϕ+ (z) ,

n=N

and therefore A−1 − (z)ϕ− (z) =

M

ζ n−κ Hn A+ (z)ϕ+ (z) ,

z ∈ Γ.

n=N

Since n − κ ≥ n − N + 1 ≥ 1 for n ≥ N , the two sides of this relation define a continuous function on C ∪ {∞} which is equal to zero for z = 0. This function is holomorphic outside Γ and, hence, by Theorem 1.5.4, holomorphic on C ∪ {∞}. Therefore, by Liouville’s theorem, it is identically zero. In particular A−1 − ϕ− ≡ 0. Since the values of A− are invertible, it follows that ϕ− ≡ 0 and, consequently, ϕ+ ≡ 0. We prove part (ii). First let κ ≥ κ+ and z ∈ D− ∪ {∞} be given. For n ≥ N , then n + κ ≥ n − N ≥ 0. Therefore, setting ϕ+ (ζ) = ζ κ T+−1 (ζ)S(ζ)T−−1 (z)v ,

ζ ∈ D+ ,

we get a function ϕ+ ∈ OE D+ . Moreover, setting ϕ− (ζ) = T− (ζ)T−−1 (z)v ,

ζ ∈ D− ∪ {∞},

we get a function ϕ− ∈ OE D− ∪ {∞} . From (7.9.10) it follows that A(ζ)ϕ+ (ζ) = T− (ζ)S −1 (ζ)T+ (ζ)ζ κ T+−1 (ζ)S(ζ)T−−1 (z)v = ζ κ ϕ− (ζ)

254


for ζ ∈ Γ. Hence (ϕ− , ϕ+ ) ∈ MA (κ, A). It remains to observe that ϕ− (z) = T− (z)T−−1 (z)v = v. Now let κ ≥ κ+ and z ∈ D+ . For n ≤ M , then n − κ ≤ M − κ+ ≤ 0. Therefore, setting ϕ− (ζ) = ζ −κ A− (ζ)H(ζ)A+ (z)v ,

ζ ∈ D− ,

we get a function ϕ− ∈ OE D− ∪ {∞} . Moreover, setting ϕ+ (ζ) = A−1 + (ζ)A+ (z)v ,

ζ ∈ D+ ,

we get a function ϕ+ ∈ OE D+ . From (7.9.9) it follows that κ A(ζ)ϕ+ (ζ) = A− (ζ)H(ζ)A+ (ζ)A−1 + (ζ)A+ (z)v = ζ ϕ− (ζ)

for ζ ∈ Γ. Hence (ϕ− , ϕ+ ) ∈ MA (κ, A). It remains to observe that ϕ+ (z) = A−1 + (z)A+ (z)v = v. : Γ → GL(E) be two continuous functions, which are equivalent 7.9.5. Let A, A with respect to Γ and GL(E) (Def. 7.1.3) and which admit local factorizations with respect to Γ (Def. 7.1.3), and let A− ∈ OGL(E) D− ∪{∞} and A+ ∈ OGL(E) D+ = A− AA+ on Γ. Then it is clear that: be functions (which then exist) such that A (ϕ− , ϕ+ ) is a (Γ, κ)-section of A, if and only if, A− ϕ− , A−1 + ϕ+ is a (Γ, κ) section of A. Hence = A− (z)M− (z, κ, Γ, A) for all z ∈ D− ∪ {∞} and κ ∈ Z, M− (z, κ, ΓA) = A−1 (z)M+ (z, κ, Γ, A) for all z ∈ D+ and κ ∈ Z. M+ (z, κ, Γ, A) + 7.9.6 Definition. Let A : Γ → GL(E) be a continuous function. An integer κ will be called a partial index of A relative to Γ or simply a partial index of A if there exists a point z ∈ C ∪ {∞} such that: if z ∈ D− ∪ {∞}, then M− (z, κ − 1, Γ, A) ⊂ M− (z, κ, Γ, A) ,

=

if z ∈ D+ , then M+ (z, κ − 1, Γ, A) ⊂ M+ (z, κ, Γ, A) .

=

Note that if A admits local factorizations relative to Γ, then, by Lemma 7.9.4, the set of partial indices of A is not empty and finite. 7.9.7 Theorem. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ (Def. 7.1.3), and let k1 > . . . > kn be the partial indices of A with respect to Γ. Then:


255

(i) For all z ∈ D− ∪ {∞}, M− (z, k1 , Γ, A) = E

and

M− (z, kn − 1, Γ, A) = {0} ,

and

M+ (z, kn − 1, Γ, A) = {0} .

and, for all z ∈ D+ , M+ (z, k1 , Γ, A) = E

(ii) For each partial index kj , 1 ≤ j ≤ n, there exists dj ∈ N∗ ∪ {∞} such that # dim M− (z, kj , Γ, A) M− (z, kj − 1, Γ, A) if z ∈ D− ∪ {∞}, # dj = dim M+ (z, kj , Γ, A) M+ (z, kj − 1, Γ, A) if z ∈ D+ . + (iii) Let 1 ≤ j ≤ n, let d ∈ N∗ , and let (ϕ− ν , ϕν ) ∈ M (kj , Γ, A), 1 ≤ ν ≤ d. Assume, for at least one point z ∈ C ∪ {∞}, the following condition C(z) is satisfied: ⎧ If z ∈ D− ∪ {∞}, then the classes in the factor space ⎪ ⎪ ⎪ # ⎪ ⎪ M− (z, kj , Γ, A) M− (z, kj − 1, Γ, A) , defined by the vectors ⎪ ⎪ ⎪ ⎨ϕ− (z), . . . , ϕ− (z) , are linearly independent. 1 d C(z) : ⎪ If z ∈ D , then the classes in the factor space + ⎪ ⎪ # ⎪ ⎪ ⎪ M+ (z, kj , Γ, A) M+ (z, kj − 1, Γ, A) , defined by the vectors ⎪ ⎪ ⎩ + ϕ1 (z), . . . , ϕ+ d (z) , are linearly independent.

Then condition C(z) is satisfied for all z ∈ C ∪ {∞}. Before proving this theorem, we use it for the following 7.9.8 Definition. With the notations from the preceding theorem we define: dj will be called the multiplicity of kj (as a partial index of A with respect to Γ), 1 ≤ j ≤ n. The family of families of subspaces M+ (z, kj , Γ, A) and M− (z, kj , Γ, A) , 1 ≤ j ≤ n, z∈D +

z∈D − ∪{∞}

will be called the filtration of A relative to Γ or simply the filtration of A. : Γ → GL(E) be two continuous functions which are 7.9.9 Remark. Let A, A equivalent relative to Γ and GL(E) (Def. 7.1.3), and which admit local factorizations with respect to Γ (Def. 7.1.3). Then it follows from Section 7.9.5 that A and have the same partial indices with the same multiplicities relative to Γ. A Proof of Theorem 7.9.7. Clearly, (ii) follows from (iii). Moreover, taking into account Lemma 7.9.4, assertion (i) follows from (ii). Therefore, it is sufficient to prove (iii). By Theorem 7.4.2, A is equivalent with respect to Γ and GL(E) to a

256


holomorphic function in a neighborhood of Γ. Therefore and by Section 7.9.5, we may assume that A is holomorphic in a neighborhood of Γ. To prove (iii), we assume that (iii) is not true, i.e., we assume that there are two points z, w ∈ C ∪ {∞} such that C(z) is satisfied, but C(w) is not satisfied. Since C(w) is not true, we can find a non-zero vector (λ1 , . . . , λd ) of complex numbers such that: d

λν ϕ− ν (w) ∈ M− (w, kj − 1, A)

if w ∈ D− ∪ {∞} ,

λν ϕ+ ν (w) ∈ M+ (w, kj − 1, A)

if w ∈ D+ .

ν=1 d ν=1

By definition of M± (w, kj − 1, A), then there exists a (kj − 1)-section (ω − , ω + ) of A such that ω − (w) =

d

λν ϕ− ν (w)

if w ∈ D− ,

λ ν ϕ+ ν (w)

if w ∈ D+ .

ν=1 +

ω (w) =

d ν=1

As observed in 7.9.3, this (kj − 1)-section can be modified so that we get a kj section ( ω− , ω + ) of A such that still ω − (w) =

d

λ ν ϕ− ν (w)

if w ∈ D− ∪ {∞} ,

ν=1

ω + (w) =

d

(7.9.11) λν ϕ+ ν (w)

if w ∈ D+ ,

ν=1

but ω − (ζ) ∈ M− (ζ, kj − 1, A)

for all ζ ∈ D− ∪ {∞} ,

ω (ζ) ∈ M− (ζ, kj − 1, A)

for all ζ ∈ D+ .

+

If w = ∞, then we set d ζ ω − (ζ) − λ ν ϕ− (ζ) ν ζ −w ν=1

for ζ ∈ D− \ {w},

d 1 + + ω (ζ) − λν ϕν (ζ) ψ+ (ζ) = ζ −w ν=1

for ζ ∈ D+ \ {w} .

ψ− (ζ) = and

(7.9.12)


257

If w = ∞, then we set

−

−

(ζ) − ψ (ζ) = ζ ω

d

λ ν ϕ− ν (ζ)

for ζ ∈ D− ,

ν=1

and ψ + (ζ) = ω + (ζ) −

d

λ ν ϕ+ ν (ζ)

for ζ ∈ D+ .

ν=1

Since A is holomorphic in a neighborhood of Γ and therefore the functions ω ± and ± ϕ ν are holomorphic in a neighborhood of Γ, then, by (7.9.11), we obtain a pair of holomorphic functions ψ − : D− ∪ {∞} → E and ψ + : D+ → E with A(ζ)ψ + (ζ) = ζ kj −1 ψ − (ζ)

for ζ ∈ Γ.

Hence (ψ − , ψ + ) is a (kj − 1)-section of A, and therefore ψ − (z) ∈ M− (z, kj − 1, A)

if z ∈ D− ∪ {∞} ,

ψ (z) ∈ M+ (z, kj − 1, A)

if z ∈ D+ .

+

(7.9.13)

On the other hand, ψ − (z) =

d

z − λμ ϕ− (z) ω (z) − z−w μ=1

− (z) − ψ − (z) = ω

d

λμ ϕ− (z)

if w = ∞ and z ∈ D− , if w = ∞ and z = ∞ ,

μ=1

d z + + ω (z) − λμ ϕ (z) ψ (z) = z−w μ=1 +

d

− (z) − λμ ϕ− (z) ψ − (z) = z ω

if w = ∞ and z ∈ D+ , if w = ∞ and z ∈ D− ,

μ=1

+ (z) − ψ + (z) = ω

d

λμ ϕ+ (z)

if w = ∞ and z ∈ D+ .

μ=1

By (7.9.12) and condition C(z), this implies that ψ − (z) ∈ M− (z, kj , A) \ M− (z, kj − 1, A)

if z ∈ D− ∪ {∞} ,

ψ (z) ∈ M+ (z, kj , A) \ M+ (z, kj − 1, A)

if z ∈ D+ ,

+

which contradicts (7.9.13).

258


Theorem 7.9.7 contains interesting information concerning the RiemannHilbert boundary problem. To state it, we first give the following 7.9.10 Definition. Let A : Γ → GL(E) be a continuous function, let z ∈ C ∪ {∞}, let v ∈ E, let κ ∈ Z, and let (ϕ+ , ϕ− ) be a κ-section of A (Def. 7.9.1). We shall say that (ϕ+ , ϕ− ) is a (Γ, κ)-section of simply a κ-section of A through (z, v) if 3 ϕ− (z) = v if z ∈ D− ∪ {∞}

and

ϕ+ (z) = v if z ∈ D+ .

With this definition, from Theorem 7.9.7 we immediately obtain: 7.9.11 Corollary (to Theorem 7.9.7). Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ, let k1 > . . . > kn be the partial indices of A (Def. 7.9.6), and let dj be the multiplicity of kj as a partial index of A (Def. 7.9.8), 1 ≤ j ≤ n. Then, for each point z ∈ C ∪ {∞}, there exist (possibly not closed) linear subspaces E1 , . . . , En of E such that E is the algebraically direct sum of E1 , . . . , En , dim Ej = dj for 1 ≤ j ≤ n and such that, for each 1 ≤ j ≤ n and each v ∈ Ej with v = 0 the following hold: (i) There exists a kj -section of A through (z, v). (ii) If μ < kj , then there exists no μ-section of A through (z, v). (iii) If (ϕ− , ϕ+ ) is a kj -section of A through (z, v), then ϕ− (ζ) ∈ M− (ζ, kj , A) \ M− (z, kj − 1, A)

for all ζ ∈ D− ∪ {∞} ,

ϕ+ (ζ) ∈ M+ (ζ, kj , A) \ M+ (z, kj − 1, A)

for all ζ ∈ D+ .

(iv) If (ϕ− , ϕ+ ) and (ψ− , ψ+ ) are two kj -section of A through (z, v), then ϕ− (ζ) − ψ− (ζ) ∈ M− (ζ, kj − 1, A)

for all ζ ∈ D− ∪ {∞} ,

ϕ+ (ζ) − ψ− (ζ) ∈ M+ (z, kj − 1, A)

for all ζ ∈ D+ .

Proof. Let z ∈ C ∪ {∞} be given. Put M− (z, κ, A) M (z, κ, A) = M+ (z, κ, A)

if z ∈ D− ∪ {∞} , if z ∈ D+ ,

κ ∈ Z.

The space En is uniquely determined. We have to set En = M (z, kn , A). If 1 ≤ j ≤ n − 1, then for Ej we can (and have to) choose an arbitrary algebraic complement of M (z, kj − 1, A) in Mj (z, kj , A). 3 For

z ∈ Γ we have to make a choice. Just as well we could require that ϕ+ (z) = v.

7.10. A general criterion for the existence of factorizations

7.10

259

A general criterion for the existence of factorizations

In this section, E is a Banach space, D+ ⊆ C is a bounded, connected, open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ , and D− = C \ D+ . 7.10.1 Theorem. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ. Then the following are equivalent: (i) A admits a canonical factorization with respect to Γ. (ii) Zero is the only partial index of A with respect to Γ (Def. 7.9.6). Proof. (i)⇒(ii): Assume that A = A− A+ is a canonical factorization of A with respect to Γ. Let (ϕ− , ϕ+ ) be a −1-section of A. Then 1 −1 A (z)ϕ− (z) = A+ (z)ϕ+ (z) z −

for all z ∈ Γ.

Then, by Theorem 1.5.4, the two sides of this equation define a holomorphic function on C ∪ {∞} which vanishes at z = ∞. Hence, by Liouville’s theorem, this function is identically zero. It follows that A−1 − ϕ− ≡ 0 on D − ∪{∞} and A+ ϕ+ ≡ 0 and A are invertible, this implies that ϕ− ≡ 0 on on D+ . Since functions A−1 + − D− ∪ {∞} and ϕ+ ≡ 0 on D+ . Hence M (−1, A) = {0}.

(7.10.1)

If z ∈ D− ∪ {∞} and v ∈ E, then, setting ϕ− (ζ) = A− (ζ)A−1 − (z)v ϕ+ (ζ) =

−1 A−1 + (ζ)A− (z)v

for ζ ∈ D− ∪ {∞} , for ζ ∈ D+ ,

we get a 0-section (ϕ− , ϕ+ ) of A with ϕ− (z) = v. Hence M (z, 0, A) = E

for all z ∈ D− ∪ {∞}.

(7.10.2)

If z ∈ D+ and v ∈ E, then, setting ϕ− (ζ) = A− (ζ)A+ (z)v ϕ+ (ζ) =

A−1 + (ζ)A+ (z)v

for ζ ∈ D− ∪ {∞} for ζ ∈ D+ ,

we get a 0-section (ϕ− , ϕ+ ) of A with ϕ+ (z) = v. Hence M (z, 0, A) = {0} = E

for all z ∈ D+ .

(7.10.3)

From (7.10.1)–(7.10.3) it follows that zero is the only partial index of A with respect to Γ.

260


(ii)⇒(i): Assume that zero is the only partial index of A. By Theorem 7.9.6 this means that M (−1, A) = {0} (7.10.4) and M− (z, 0, A) = E

for all z ∈ D− ∪ {∞} ,

M+ (z, 0, A) = E

for all z ∈ D+ .

(7.10.5)

For each ϕ = (ϕ− , ϕ+ ) ∈ M (0, A), we define Φ− (z)ϕ = ϕ− (z)

for z ∈ D− ∪ {∞} ,

Φ+ (z)ϕ = ϕ+ (z)

for z ∈ D+ .

Recall that we consider M (0, A) as a Banach space (cf. Def. 7.9.1). Then it follows from the maximum principle that

for all z ∈ D− ∪ {∞} , Φ− (z) ∈ L M (0, Γ, A), E

Φ+ (z) ∈ L M (0, Γ, A), E for all z ∈ D+ . By Theorem 7.9.7, the operators Φ− (z), z ∈ D− ∪ {∞}, and Φ+ (z), z ∈ D+ , are injective. Moreover, by (7.10.5), these operators are surjective. Hence, by the Banach open mapping theorem, they are invertible from M (0, A) onto E. By Theorem 7.4.2 and Remark 7.9.9, in this proof, we may assume that A is holomorphic in a neighborhood of Γ. Then, for each ϕ = (ϕ− , ϕ+ ) ∈ M (0, A), it follows from the relation ϕ− (z) = A(z)ϕ+ (z) ,

z ∈ Γ,

that Φ− ϕ = ϕ− is holomorphic on D− ∪ {∞}, and Φ+ ϕ = ϕ+ is holomorphic on D+ . Hence the functions

Φ− : D− ∪ {∞} → L M (0, A), E ,

Φ+ : D+ → L M (0, A), E are holomorphic (Theorem 1.7.1). Now we fix some point z0 ∈ Γ, and set A− (z) = Φ− (z)Φ−1 − (z0 ) A+ (z) =

Φ+ (z)Φ−1 − (z0 )

for z ∈ D− ∪ {∞} , for z ∈ D+ .

Then A = A− A−1 + on Γ. Indeed, let v ∈ E and z ∈ Γ be given. Set ϕ = (ϕ− , ϕ+ ) = Φ−1 + (z)v.


261

Then ϕ− (z) = A(z)ϕ+ (z) and Φ+ (z)ϕ = v, and therefore −1 −1 A− (z)A−1 + (z)v = Φ− (z)Φ− (z0 )Φ− (z0 )Φ+ (z)v = Φ− (z)ϕ = ϕ− (z) = A(z)ϕ+ (z) = A(z)Φ+ (z)ϕ = A(z)v .

7.10.2 Lemma. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ. Suppose A has negative partial indices with respect to Γ (Def. 7.9.6). Denote these negative partial indices by κ1 > . . . > κm . Let N be the subspace of M (0, A) (Def. 7.9.1) which consist of the sections (ϕ− , ϕ+ ) ∈ M (0, A) with ϕ− (z) ∈ M− (z, κ1 , A)

for all z ∈ D− ∪ {∞} ,

ϕ+ (z) ∈ M+ (z, κ1 , A)

for all z ∈ D+ .

If each of the negative partial indices κ1 , . . . , κm has finite multiplicity (Def. 7.9.8), then N is finite dimensional. Proof. Let dj be the multiplicity of κj , 1 ≤ j ≤ m. Set κ0 = 0 and κm+1 = κm − 1. Suppose dj < ∞ for 1 ≤ j ≤ m. Then we can find vectors vj,1 , . . . , vj,dj ∈ M (0, κj , A), 1 ≤ j ≤ m, which define a basis in the factor space 3 M (0, κj , A) M (0, κj+1 , A) . Further, for 1 ≤ j ≤ m and 1 ≤ ν ≤ dj , we choose a section + ϕj,ν = ϕ− j,ν , ϕj,ν ∈ M (κj , A) with ϕ+ j,ν (0) = vj,ν . By Theorem 7.9.7, then, for 1 ≤ j ≤ m, − for all z ∈ D− ∪ {∞} , the vectors ϕ− j,1 (z), . . . , ϕj,dj (z) # define a basis in the factor space M− (z, κj , A) M− (z, κj+1 , A) , + and, for all z ∈ D+ , the vectors ϕ+ j,1 (z), . . . , ϕj,dj (z) # define a basis in the factor space M+ (z, κj , A) M+ (z, κj+1 , A) .

(7.10.6)

Note that this in particular implies that for all z ∈ D− ∪ {∞} , the vectors ϕ− j,ν (z) , 1 ≤ j ≤ m, 1 ≤ ν ≤ dj , form a basis of the space M− (z, κ1 , A), and, for all z ∈ D+ , the vectors ϕ+ j,ν (z), 1 ≤ j ≤ m, 1 ≤ ν ≤ dj , form a basis in the space M+ (z, κ1 , A) . Moreover, for 1 ≤ j ≤ m, 1 ≤ ν ≤ dj and 0 ≤ s ≤ −κj , we define κj +s − ϕj,ν (z) ϕ− s,j,ν (z) = z

ϕ+ s,j,ν (z)

=z

s

ϕ+ j,ν (z)

for z ∈ D− , for z ∈ D+ .

(7.10.7)

262


+ Then each ϕ− s,j,ν ϕs,j,ν belongs to the space N . To prove that dim N < ∞, now it is sufficient to prove that the finite system ϕs,j,ν 1≤j≤m,1≤ν≤dj ,0≤s≤−κj generates N . For that we consider an arbitrary (φ− , φ+ ) ∈ N . By (7.10.7), there are + uniquely determined continuous functions λ− j,ν : D − ∪ {∞} → C and λj,ν : D + → C, which are holomorphic in D− ∪ {∞} and D+ , respectively, such that − λ− φ− (z) = j,ν (z)ϕj,ν (z) for z ∈ D − ∪ {∞} , 1≤j≤m,1≤ν≤dj

φ+ (z) =

+ λ+ j,ν (z)ϕj,ν (z)

(7.10.8)

for z ∈ D+ .

1≤j≤m,1≤ν≤dj + Since φ− (z) = A(z)φ+ (z) and z κj ϕ− j,ν (z) = A(z)ϕj,ν (z) for z ∈ Γ, this implies that − − λ− z κj λ+ j,ν (z)ϕj,ν (z) = j,ν (z)ϕj,ν (z) for z ∈ Γ . 1≤j≤m,1≤ν≤dj

1≤j≤m,1≤ν≤dj

By (7.10.7) this implies that κj + λ− j,ν (z) = z λj,ν (z)

for z ∈ Γ , 1 ≤ j ≤ m, 1 ≤ ν ≤ dj .

It follows by Liouville’s theorem that, for some numbers αs,j,ν , λ− j,ν (z) =

−κj

−κj

αs,j,ν z κj +s

and λ+ j,ν (z) =

s=0

αs,j,ν z s .

s=0

Together with (7.10.8) this implies φ− =

αs,j,ν ϕ− s,j,ν ,

1≤j≤m,1≤ν≤dj ,0≤s≤−κj

φ+ =

1≤j≤m,1≤ν≤dj ,0≤s≤−κj

αs,j,ν ϕ+ s,j,ν .

7.10.3 Theorem. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ. Then the following are equivalent: (i) A admits a factorization with respect to Γ. (ii) If A has non-zero partial indices (Def. 7.9.6), then each of them has finite multiplicity (Def. 7.9.8).


263

Moreover: If these equivalent conditions are satisfied and if A = A− ΔA+ is a factorization of A with respect to Γ, then: (a) If zero is the only partial index of A with respect to Γ, then Δ ≡ I. (b) If there exist non-zero partial indices of A with respect to Γ, then: – The numbers κ1 > . . . > κn from Definition 7.1.1 are the non-zero partial indices of A. – If P1 , . . . , Pn are the projections from Definition 7.1.1, the dim Pj is the multiplicity of κj (as a partial index of A), 1 ≤ j ≤ n. - If zero is a partial index of A (which is always the case for dim E = ∞), then dim P0 is the multiplicity of zero (as a partial index of A). Proof. First assume that there exists a factorization A = A− ΔA+ of A with respect to Γ. Then, by Section 7.9.5, the diagonal factor Δ and A have the same partial indices with the same multiplicities. If zero is the only non-zero partial index of Δ, then it is clear that Δ ≡ I. If there exist non-zero partial indices of Δ, then it is also clear that, with the notation from Definition 7.1.1, the numbers κ1 > . . . > κn are these non-zero partial indices with the multiplicities dim P1 , . . . , dim Pn , respectively. Further it is clear that zero is a partial index of Δ, if and only if, P0 = 0 and that then dim P0 is the multiplicity of zero. Now we assume that condition (ii) is satisfied. It remains to prove that then A admits a factorization. If zero is the only partial index of A, we know this from Theorem 7.10.1. Assume there exist non-zero partial indices of A. For simplicity we consider only the case when there exist both positive and negative partial indices. (It is clear how to modify the proof in the other cases.) Let κ1 > . . . > κm be the positive partial indices of A, and let κm+1 > . . . > κn be the negative partial indices of A. Let N be the subspace of M (0, A) defined in Lemma 7.10.2. By this lemma, N is finite dimensional. If (ϕ− , ϕ+ ) ∈ M (0, A), then we define

for z ∈ D− ∪ {∞} , Φ− (z) (ϕ− , ϕ+ ) = ϕ− (z)

Φ+ (z) (ϕ− , ϕ+ ) = ϕ+ (z) for z ∈ D+ . It follows from the maximum principle for holomorphic functions that in this way functions

Φ− : D− ∪ {∞} → L M (0, A), E ,

(7.10.9) Φ+ : D+ → L M (0, A), E are defined.

264


By Theorem 7.4.2 and Remark 7.9.9 we may assume that A is holomorphic in a neighborhood of Γ. Then, for each ϕ = (ϕ− , ϕ+ ) ∈ M (0, A), it follows from z ∈ Γ,

ϕ− (z) = A(z)ϕ+ (z) ,

that Φ− ϕ = ϕ− is holomorphic on D− ∪ {∞}, and Φ+ ϕ = ϕ+ is holomorphic on D+ . Hence the functions (7.10.9) are holomorphic (Theorem 1.7.1). Note that, by definition, M− (z, 0, A) = Im Φ− (z)

for z ∈ D− ∪ {∞} ,

M+ (z, 0, A) = Im Φ+ (z)

for z ∈ D+

(7.10.10)

and for z ∈ Γ .

Φ− (z) = A(z)Φ+ (z)

(7.10.11)

Since the positive partial indices of A have finite multiplicities, it follows from Theorem 7.9.7 that the spaces M− (z, 0, A), z ∈ D− ∪ {∞}, and M+ (z, 0, A), z ∈ D+ , are of finite codimension in E. Since the operators Φ− (z) and Φ+ (z) are linear and bounded, this implies together with (7.10.10) that these spaces are closed in E. Since N is finite dimensional, N is complemented in M (0, A). Therefore, M (0, A) can be written as a direct sum .

M (0, A) = N + N ⊥ with some closed subspace N ⊥ of M (0, A). Set E0− (z) = Φ− (z)N ⊥ E0+ (z)

= Φ+ (z)N

⊥

for z ∈ D− ∪ {∞} , for z ∈ D+ .

(7.10.12)

Then, by definition of N and by Theorem 7.9.7, Φ− (z)N ⊆ M− (z, κm+1 , A) ,

z ∈ D− ∪ {∞} ,

Φ+ (z)N ⊆ M+ (z, κm+1 , A) ,

z ∈ D+ ,

E0− (z) E0+ (z)

∩ M− (z, κm+1 , A) = {0} ,

z ∈ D− ∪ {∞} ,

∩ M+ (z, κm+1 , A) = {0} ,

z ∈ D+ .

Since the spaces M± (z, 0, A) are closed and the spaces M± (z, κm+1 , A) are finite dimensional, this together with (7.10.10) implies that also the spaces E0− (z), z ∈ D− ∪ {∞}, and E0+ (z), z ∈ D+ , are closed and that .

M− (z, 0, A) = E0− (z) + M− (z, κm+1 , A) , z ∈ D− ∪ {∞} , .

M+ (z, 0, A) = E0+ (z) + M+ (z, κm+1 , A) , z ∈ D+ .

(7.10.13)


265

Moreover, by Theorem 7.9.7 and the definition of N , Ker Φ− (z) ∩ N ⊥ = {0} , Ker Φ+ (z) ∩ N

⊥

z ∈ D− ∪ {∞} ,

= {0} ,

z ∈ D+ .

(7.10.14)

Hence, by the Banach open mapping theorem, for z ∈ D− ∪ {∞}, the operator Φ− (z)N ⊥ is invertible as an operator from N ⊥ to E0− (z), and, for z ∈ D+ , the operator Φ+ (z)N ⊥ is invertible as an operator from N ⊥ to E0+ (z). Now we fix some point z0 ∈ Γ. Let Φ− (z0 ) : E0− (z0 ) → N ⊥ be the inverse (−1) of Φ− (z0 ) as an operator from N ⊥ to E0− (z0 ), and let Φ+ (z0 ) : E0+ (z0 ) → N ⊥ + ⊥ be the inverse of Φ+ (z0 ) as an operator from N to E0 (z0 ). Then, setting (−1)

A− 0 (z) = Φ− (z)Φ−

(−1)

A+ 0 (z)

=

for z ∈ D− ∪ {∞} ,

(z0 )

(−1) Φ+ (z)Φ+ (z0 )A−1 (z0 )

for z ∈ D+ ,

we obtain holomorphic functions − A− 0 : D − ∪ {∞} −→ L E0 (z0 ), E , − A+ 0 : D + −→ L E0 (z0 ), E , with injective values and such that − − A− 0 (z)E0 (z0 ) = E0 (z)

forz ∈ D− ∪ {∞} ,

− A+ 0 (z)E0 (z0 )

for z ∈ D+ .

=

E0+ (z)

(7.10.15)

Note that, by (7.10.11), for z ∈ Γ, (−1)

A(z)A+ 0 (z) = A(z)Φ+ (z)Φ+ (−1)

Since, again by (7.10.11), Φ+

(z0 )A−1 (z0 ) = Φ− (z)Φ+

(−1)

(z0 )A−1 (z0 ) = Φ−

(−1)

− A(z)A+ 0 (z) = A0 (z)

Set

(z0 )A−1 (z0 ).

(z0 ), this implies that

for z ∈ Γ .

En− (z0 ) = M− (z0 , κn , A)

and choose, for 1 ≤ j ≤ n − 1, a subspace

Ej− (z0 )

(7.10.16) (7.10.17)

of M (z0 , κj , A) such that

.

M− (z0 , κj , A) = Ej− (z0 ) + M− (z0 , κj+1 , A)

if j = m ,

.

− (z0 ) + M− (z0 , 0, A) . M− (z0 , κm , A) = Em

(7.10.18)

From (7.10.13), (7.10.17) and (7.10.18) it follows that .

.

.

E = E0− (z0 ) + E1− (z0 ) + . . . + En− (z0 ) .

(7.10.19)

266


By (7.10.17) and (7.10.18), for 1 ≤ j ≤ n, we can find sections

− + + , ϕ 1 ≤ j ≤ n, ϕj1 , ϕj1 , . . . , ϕ− jdj jdj ∈ M (z0 , κj , A) , − − − such that ϕ− j1 (z0 ), . . . , ϕjdj (z0 ) is a basis of Ej (z0 ). Let Ej (z), z ∈ D − ∪ {∞},

j− + be the space spanned by ϕj− 1 (z), . . . , ϕdj (z), and let Ej (z), z ∈ D + , be the space + spanned by ϕ+ 1j (z), . . . , ϕjdj (z). Then, from Theorem 7.9.7 and (7.10.19) it follows that .

.

.

.

.

E = E0− (z) + E1− (z) + . . . + En− (z) , E=

E0+ (z)

Now, for 1 ≤ j ≤ n, let

.

+

E1+ (z)

+ ... +

z ∈ D− ∪ {∞} , z ∈ D+ .

En+ (z) ,

− A− j : D − ∪ {∞} −→ L Ej (z0 ), E

be the holomorphic function defined by − − A− j (z)ϕjν (z0 ) = ϕjν (z) ,

and let

z ∈ D− ∪ {∞} ,

1 ≤ ν ≤ dj ,

− A+ j : D + −→ L Ej (z0 ), E

be the holomorphic function defined by − + A+ j (z)ϕjν (z0 ) = ϕjν (z) ,

z ∈ D+ ,

1 ≤ ν ≤ dj .

Note that then − − A− j (z)Ej (z0 ) = Ej (z) ,

z ∈ D− ∪ {∞} ,

− A+ j (z)Ej (z0 )

z ∈ D+ ,

=

Ej+ (z) ,

1 ≤ j ≤ n,

1 ≤ j ≤ n.

(7.10.20)

Moreover, then, for z ∈ Γ, 1 ≤ j ≤ n and 1 ≤ ν ≤ dj , − + − κj − κj − A(z)A+ j (z)ϕjν (z0 ) = A(z)ϕjν (z) = z ϕjν (z) = z Aj (z)ϕjν (z0 ) .

Hence

+ z κj A− j (z) = A(z)Aj (z) ,

z ∈ Γ,

1 ≤ j ≤ n.

(7.10.21)

By (7.10.19) there are mutually disjoint projectors Pj , 0 ≤ j ≤ n, in E with I = P0 + P1 + . . . + Pn

and

Im Pj = Ej− (z0 ) ,

0 ≤ j ≤ n.

Set A− (z) = A+ (z) =

n j=0 n

A− j (z)Pj

for z ∈ D− ∪ {∞} ,

A+ j (z)Pj

for z ∈ D+ ,

j=0

Δ(z) = P0 +

n j=1

z κj Pj .

7.11. Comments

267

By (7.10.19), (7.10.15) and (7.10.20), in this way holomorphic functions A− : D− ∪ {∞} −→ GL(E)

and A+ : D− −→ GL(E)

are defined. It follows from (7.10.21) and (7.10.16) that A(z)A+ (z) = A− (z)Δ(z)

for z ∈ Γ .

Hence A = A− ΔA−1 + is a factorization of A with respect to Γ.

7.11

Comments

This type of factorization written without the diagonal factor for matrix functions was considered for the first time in the pioneering work of Plemelj [Pl]. The complete proofs were given by N.I. Muschelishvili [Mu]. This was the motivation behind the term Plemelj-Muschelishvili factorisation. This form of the factorization which is considered here was first proposed for matrix functions in [GK]. In some sources, this form of factorization is called Gohberg-Krein factorization or Wiener-Hopf factorization or factorization along a contour. For connections with Wiener-Hopf and T¨ oplitz operators, see the next chapter. The example (8.13.2) was given in [GK]. A general review for factorizations in algebras and applications can be found in [Go4]. The first factorization theorem for operator-valued functions (the result of Section 7.8) was proved in [Go2]. Another proof of this result was given in [Le1, Le3] (see also [CG]). The local principle for matrix functions was proposed by M.A. Shubin [Sh1]. The extension of this principle to operator functions as well as applications one can find in [GL4]. For other directions of developments in matrix and operator functions, see [BGK, GKS, GLR, GGK1, CG]. Factorization theory of operator-valued functions represents also an important tool in spectral theory of operator functions and operator polynomials (see, for instance, [Ma] and [Ro]).

Chapter 8

Wiener-Hopf operators, Toeplitz operators and factorization In this chapter we continue to study the factorization problem, where now the emphasis is on the connection with Wiener-Hopf and Toeplitz operators. This chapter also contains applications to operator-valued Wiener-Hopf equations and equations with infinite T¨ oplitz matrices. The local principle continues to play an important role.

8.1

Holomorphic operator functions

In this section E is a Banach space, D+ ⊆ C is a bounded connected open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ , and D− := C\D+ . Further, we assume that a bounded neighborhood W of Γ is fixed such that each connected component of W contains at least one connected component of Γ.1 We set W+ = D+ ∪ W and W− = D− ∪ W . 8.1.1 Definition. We denote by OE (W ) the Banach space of continuous E-valued functions, which are holomorphic in W , endowed with the norm

f W := max f (z) , z∈W

f ∈ OE (W ) .

Further, let OE (W + ) be the subspace of all functions in OE (W ) which admit a holomorphic extension to W+ , and let OE 0 (W − ∪ {∞}) be the subspace of all 1 It

is possible but not necessary that W is a small neighborhood of Γ. An important example is also W = z ∈ C r < |z| < R with 0 < r < R < ∞ such that Γ ⊆ W .

270

Chapter 8. Wiener-Hopf operators, Toeplitz operators

functions f ∈ OE (W ) which admit a holomorphic extension to W− ∪ (∞} with f (∞) = 0. Since each connected component of W contains at least one connected component of Γ, it follows from Theorem 3.7.3 and Proposition 3.1.3 that OE (W ) is the direct sum of OE (W + ) and OE 0 (W − ∪ {∞}). We denote by P the projector from OE (W ) onto OE (W + ) parallel to E E E OE 0 (W − ∪{∞}). As O (W + ) and O 0 (W − ∪{∞}) are closed subspaces of O (W ), E this projector is continuous with respect to the topology of O (W ) (by the Banach open mapping theorem). We set Q = I − P. 8.1.2 Definition. Let A : W → L(E) be a continuous function which is holomorphic in W , and let MA : OE (W ) −→ OE (W ) be the operator of multiplication by A: f ∈ OE (W ) ,

(MA f )(z) = A(z)f (z) ,

z ∈ W.

Then we denote by WA the bounded linear operator on OE (W + ) defined by f ∈ OE (W + ) . (WA )f = P MA f , This operator WA will be called the Wiener-Hopf operator defined by A on OE (W + ). 8.1.3 Lemma. Let A, B : W → GL(E) be continuous functions which are holomorphic in W , and let WA and WB be the Wiener-Hopf operators defined in OE (W + ) by A and B respectively. Suppose A and B are equivalent with respect to Γ (Def. 7.1.3). Then WA and WB are equivalent.2 Proof. By hypothesis there are continuous functions T− : D− ∪ {∞} → GL(E) and T+ : D+ → GL(E) which are holomorphic in D− ∪ {∞} and D+ , respectively, such that A(z) = T− (z)B(z)T+ (z) (8.1.1) for z ∈ Γ. Since A and B are continuous on W and holomorphic in W , and since each connected component of W contains at least one connected component of Γ, from this relation it follows that T− admits a continuous and invertible extension to W − ∪{∞} which is holomorphic in W− , and T+ admits a continuous and invertible extension to W + which is holomorphic in W+ . We denote these extensions also by T− and T+ . Then (8.1.1) holds for all z ∈ W . It is clear that PMT+ P = MT+ P

and

MT −1 P = MT −1 P . +

+

(8.1.2)

Also it is clear that PMT− Q = PMT −1 Q = 0, which implies that −

PMT− P = PMT− 2 Two

and

PMT −1 P = PMT −1 . −

−

(8.1.3)

operators T and S in a Banach space X are called equivalent if there exist invertible operators V, W in X such that T = V SW .

8.1. Holomorphic operator functions

271

From (8.1.2) and (8.1.3) it follows that WT− and WT+ are invertible, where = WT −1 WT−1 − −

WT−1 = WT −1 . +

and

+

From (8.1.1)–(8.1.3) it follows that WT− WB WT+ = PMT− PMB PMT+ OE (W + ) = PMT− MB MT+ E = PMA O (W + )

O E (W + )

Hence WA and WB are equivalent.

= WA .

8.1.4 Theorem. Let A : W → GL(E) be a continuous function which is holomorphic in W , and let WA be the Wiener-Hopf operator defined by A in OE (W + ). Then: (i) The function A admits a canonical factorization with respect to Γ, if and only if, WA is invertible. (ii) The function A admits a factorization with respect to Γ, if and only if, WA is a Fredholm operator. If this is the case, with the notations from Definition 7.1.1, dim Coker WA = κj dim Pj , 1≤j≤n , κj >0

dim Ker WA = −

(8.1.4) κj dim Pj ,

1≤j≤n , κj 0 resp. if there is no j with κj < 0. Proof. First assume that A admits a factorization A = A− ΔA+ . Then it is easy to see (using Theorem 1.5.4) that WΔ is a Fredholm operator, where, with the notations from Definition 7.1.1, dim Coker WΔ = κj dim Pj , 1≤j≤n , κj >0

dim Ker WΔ = −

(8.1.5) κj dim Pj .

1≤j≤n , κj 0 be a partial index of A with multiplicity d. By Corollary 7.9.11, there is a d-dimensional linear subspace V of E such that, for each vector v ∈ V with v = 0, there exists no (κ − 1)-section of A through (∞, v). Since κ − 1 ≥ 0, it follows (cf. Remark 7.9.3) that there exist no 0-section through (∞, v). Now let V be the linear subspace of OE (W + ) which consists of all constant functions with value in V . Then dim V = d .

(8.1.9)

V ∩ Im WA = {0} .

(8.1.10)

Moreover Indeed, assume for some v0 ∈ V the constant function with value v0 belongs to Im WA . Then there exist ϕ+ ∈ OE (W + ) and ϕ− ∈ OE 0 W − ∪ {∞} with A(z)ϕ+ (z) = v0 + ϕ− (z)

for z ∈ W .

Since ϕ− (∞) = 0 (by definition of OE 0 (W − ∪ {∞})), then (ϕ+ , v0 + ϕ− ) is a 0-section of A through (∞, v). Since there do not exist 0-sections of A through

8.2. Factorization of G ω (E)-valued functions

273

(∞, v) if v ∈ V and v = 0, it follows that v0 = 0. From (8.1.10) and (8.1.9) it follows that Coker WA = d.

8.2

Factorization of G ω (E)-valued functions

Here we use the same notation as in Section 7.8 and prove the result on factorization of G ω (E)-valued functions announced already there. 8.2.1 Lemma. Let 0 < r < R < ∞ such that Γ ⊆ W := z ∈ C r < |z| < R , and let V : W → F ω (E) be holomorphic. Then, with the notations introduced at the beginning of Section 8.1 and in Definition 8.1.2, the operators PMV Q and QMV P are compact as operators on OE (W ). Proof. Let ∞

V (z) =

z n Vn

n=−∞

be the Laurent expansion of V . Since V is holomorphic on W (i.e., in a neighborhood of W ), this series converges uniformly on W . It follows that PMV Q =

∞

∞

PMzn Vn Q and QMV P =

n=−∞

QMzn Vn P

n=−∞

where the sums converge with respect to the operator norm of L(OE (W )). Therefore it is sufficient to prove that for all n ∈ Z the operators PMzn Vn Q and QMzn Vn P are compact as operators on OE (W ). Let n ∈ Z be given, and let (ϕν )ν∈N be a bounded sequence in OE (W ). We have to find a subsequence ϕνj of it such that the sequences PMzn Vn Q ϕνj and

QMzn Vn P ϕνj converge in OE (W ). Let ϕν (z) =

∞

z k ϕνk ,

z∈W

k=−∞

be the Laurent expansion of ϕν . Then by Cauchy’s inequality, for each k ∈ Z, the sequence (ϕνk )ν∈N is bounded in E. Since the operator Vn is compact, then there exists a sequence ν1 ≤ ν2 ≤ . . . of numbers νj ∈ N such that, for all k ∈ Z with −n ≤ k ≤ −1

if n ≥ 1

and

0 ≤ k ≤ −n − 1

if n ≤ −1 ,

274


the subsequence (Vn ϕνj ,k )j∈N∗ of (Vn ϕν,k )ν∈N converges in E. Since

PMzn Vn Q ϕνj = 0 if n ≤ 0 ,

−1 −1

PMzn Vn Q ϕνj = Pz n Vn z k ϕνj ,k = z k+n Vn ϕνj ,k k=−∞

QMzn Vn P ϕνj = 0

if n ≥ 1 ,

k=−n

if n ≥ 0 ,

QMzn Vn P ϕνj = Qz Vn n

this implies that sequences

∞ k=0

k

z ϕνj ,k =

−n−1

z k+n Vn ϕνj ,k

if n ≤ −1 ,

k=0

PMzn Vn Q ϕνj and QMzn Vn P ϕνj converge in

OE (W ).

8.2.2 Theorem. Let A : Γ → G ω (E) be a continuous function which admits local factorizations with respect to Γ and GL(E). Then A admits a factorization with respect to Γ and G ω (E). Proof. By Theorem 7.4.2, we may assume that A is defined, holomorphic and with values in G ω on C∗ . Choose 0 < r < R < ∞ such that Γ ⊆ W := z ∈ C r < |z| < R . Let V : W → F ω (E) be the function with A = I + V . Let MA and MV be the operators of multiplication by A and V , respectively. By Lemma 8.2.1, PMV Q and QMV P are compact as operators on OE (W ). Since MA is invertible as an operator on OE (W ) and MA = PMA P + QMA Q + PMV Q + PMV Q , this implies that PMA P + QMA Q E is Fredholm as an operator on O (W ).EIn particular, the Wiener-Hopf operator WA = PMA OE (W + ) defined by A in O (W + ) is a Fredholm operator. By Theorem 8.1.4 this implies that A admits a factorization with respect to Γ and GL(E). Finally, it follows from Proposition 7.8.1 that A admits a factorization with respect to Γ and G ω (E).

Under certain additional assumptions about the function A in Theorem 8.2.2, we can say correspondingly more about the factors of the factorizations of A. We have: 8.2.3 Corollary (to Theorem 8.2.2). Let 0 < α < 1 and k ∈ N, where, for k ≥ 1, we additionally assume that Γ is of class C k (Def. 3.4.1). Let A : Γ → G ω (E) be a function of class Ck+α (Def. 3.4.3). Then:

8.2. Factorization of G ω (E)-valued functions

275

(i) A admits a factorization with respect to Γ and G ω (E). (ii) If A = A− ΔA+ is an arbitrary factorization of A with respect to Γ, then automatically, the factors A− and A+ are of class Ck+α on D− and D+ , respectively. Proof. Part (i) of Theorem 7.3.1 in particular states that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 8.2.2. Part (ii) of the corollary follows from part (ii) of Theorem 7.3.1 8.2.4 Corollary (to Theorem 8.2.2). Let Γ = T be the unit circle, and let R be a Banach algebra of continuous L(E)-valued functions satisfying conditions (A), (B) and (C) in Section 7.2.4. For example, let R = W L(E) be the Wiener algebra (see Section 7.2.1). Let A : Γ → G ω (E) be a function which belongs to R. Then: (i) The function A admits a factorization with respect to Γ and G ω (E). (ii) If A = A− ΔA+ is an arbitrary factorization of A with respect to Γ, then automatically, the factors A− and A+ belong to the algebra R. Proof. Part (ii) of Theorem 7.2.5 in particular states that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 8.2.2. Part (ii) of the corollary follows from part (iii) of Theorem 7.2.5. Theorem 8.2.2 admits the following generalization: 8.2.5 Theorem. Let A : Γ → GL(E) be a continuous admits local function which factorizations with respect to Γ and GL(E). Let G L(E)/F ω (E) be the group of invertible elements of the factor algebra L(E)/F ω (E), and let π : L(E) → L(E)/F ω (E) be the canonical map. Then the following are equivalent: (i) The function A admits a factorization with respect to Γ and GL(E). (ii) The function π(A) admits a canonical factorization with respect to Γ and G L(E)/F ω (E) . Proof. (i)⇒(ii) is clear. We prove (ii)⇒(i). Assume there is a canonical factoriza tion π(A) = f− f+ of π(A) with respect to Γ and G L(E)/F ω (E) . By Theorem 7.4.2 we may assume that A is holomorphic in a neighborhood of Γ. Then also π(A) is holomorphic in a neighborhood of Γ, and it follows from the relations −1 f− = π(A)f+

and

−1 f+ = f− π(A)

that f− is holomorphic in a neighborhood of D− ∪{∞} and f+ is holomorphic in a neighborhood of D+ . Then, by Theorem 7.8.2, we can find holomorphic functions A− : D− ∪ {∞} → GL(E) and A+ : D+ → GL(E) such that π(A− ) = f− and π(A+ ) = f+ . Setting = A−1 AA−1 , A − +

276


: Γ → GL(E). Since π(A) = f −1 π(A)f −1 = 1 we define a holomorphic function A − + on Γ, where 1 is the unit element in L(E)/F ω (E), we see that the values of A ω belong to G (E). By Theorem 7.7, A admits a factorization with respect to Γ and is equivalent to A with respect to Γ and GL(E), GL(E). Since, by definition, A this completes the proof. Note the following obvious corollary to Theorem 8.2.5 (which is not obvious at all without this theorem): 8.2.6 Corollary. Let A : Γ → GL(E) be a continuous function which admits local factorizations with respect to Γ. Suppose, there is a finite number of points p1 , . . . , pm ∈ D+ such that A admits a continuous extension to D+ \ {p1 , . . . , pm } (also denoted by A) which is holomorphic in D+ \ {p1 , . . . , pm } and has the following property: For each z ∈ D+ \ {p1 , . . . , pm }, the value A(z) is a Fredholm operator, and, for all 1 ≤ j ≤ m, the Laurent expansion of A at pj is of the form ∞

A(z) =

(z − pj )n An ,

n=−∞

where A0 is a Fredholm operator and each of the coefficients An with n ≤ 0 is compact. Then A admits a factorization with respect to Γ. Clearly, there is also a corresponding corollary with respect to D− ∪ {∞}.

8.3

The space L2 (Γ, H)

The results on factorization obtained up to now can be essentially improved if we restrict ourselves to Hilbert spaces. In the present section, we introduce a technical tool for this: the Hilbert space of Hilbert space-valued functions on a contour. Here we use the following notations: – H is a separable Hilbert space with the scalar product ·, ·H and the norm

· H . – D+ ⊆ C is a bounded connected open set with piecewise C 1 -boundary Γ. We assume that 0 ∈ D+ , and we set D− := C \ D+ . – We give Γ the orientation as the boundary of D+ , and we denote by |dz| the Euclidean volume form of Γ. – If U ⊆ C is an open set, then we denote by OH (U ) the space of continuous functions on U which are holomorphic in U , and by OH (U ) we denote the subspace of OH (U ) which consists of the functions which admit a holomorphic extension to some neighborhood of U . – C 0 (Γ, H) is the space of continuous H-valued functions on Γ.

8.3. The space L2 (Γ, H)

277

– OH (Γ) is the subspace of C 0 (Γ, H) which consists of the functions which admit a holomorphic extension to some neighborhood of Γ. – O0H (D− ) is the subspace of OH (D− ) which consists of the functions with f (∞) = 0. 8.3.1. In C 0 (Γ, H) we introduce a scalar product ·, ·L2 (Γ,H) and the corresponding norm · L2 (Γ,H) , setting

for f, g ∈ C 0 (Γ, H) f, gL2 (Γ,H) = f (z), g(z) H |dz| Γ

and

f L2 (Γ,H) =

4 f, f L2 (Γ,H)

for f ∈ C 0 (Γ, H) .

We define L2 (Γ, H) to be the completion of C 0 (Γ, H) with respect to the norm

· L2 (Γ,H) . 8.3.2. For the scalar case H = C, the space L2 (Γ, C) is usually interpreted as the space of square integrable complex-valued functions on Γ. We avoid such an interpretation in the case of an arbitrary separable Hilbert space H (although this is possible). By our definition, the elements of L2 (Γ, H) are equivalence classes of Cauchy sequences of continuous functions. Nevertheless, many operations which can be defined pointwise in the case of a true function can be applied also to the elements of L2 (Γ, H). For example, let ϕ : Γ → C be a scalar continuous function and let f ∈ L2 (Γ, H) be represented by the Cauchy sequence {fn }n∈N of functions fn ∈ C 0 (Γ, H). Then ϕf is defined to be the element in L2 (Γ, H) represented by the Cauchy sequence {ϕfn }n∈N . Note that then ϕf 2 = max ϕ(z)ϕf 2 L (Γ,H)

L (Γ,H)

z∈Γ

In the same way, for each continuous operator function A : Γ → L(H) and each f ∈ L2 (Γ, H), the product Af is well defined. Then Af 2 = max A(z)L(H) ϕf L2 (Γ,H) . L (Γ,H) z∈Γ

Moreover, denote by L (Γ, C) the Banach space of integrable complex-valued functions on Γ with the norm

f L1 (Γ,C) = |f ||dz| , f ∈ L1 (Γ, C) . 1

Γ

Then, for any two elements f, g ∈ L2 (Γ, H), a function f, gH ∈ L1 (Γ, C) is well defined such that f, gL2 (Γ,H) = f, gH |dz| . Γ

278


Indeed, let {fn }n∈N and {gn }n∈N be two sequences of functions from C 0 (Γ, H) which represent f and g, respectively. Then, by the Cauchy-Schwarz inequality in H, pointwise on Γ we have

fn , gn H − fm , gm H ≤ fn − fm H gn H + fm H gn − gm H , which implies, by the Cauchy-Schwarz inequality in L2 (Γ, C), that

fn , gn H − fm , gm H 1 L (Γ,C) ≤ fn − fm L2 (Γ,H) gn L2 (Γ,H) + fm L2 (Γ,H) gn − gm L2 (Γ,H) . Hence fn , gn H n∈N is a Cauchy sequence in L1 (Γ, C), and the function f, gH can be defined to be the limit of this sequence in L1 (Γ, C). Finally we note that, for each f ∈ L2 (Γ, H), the function 4

f H := f, f H belongs to L2 (Γ, C) and 2 f 2 L (Γ,H) =

f 2H |dz| . Γ

8.3.3. Note that, by Corollary 3.3.3, OH (Γ) is dense in C 0 (Γ, H) with respect to uniform convergence on Γ. Hence OH (Γ) is dense in L2 (Γ, H) with respect to the norm · L2 (Γ,H) . 8.3.4. The linear map IΓ from C 0 (Γ, H) to H defined by f ∈ C 0 (Γ, H) , IΓ (f ) := f (z)dz ,

(8.3.1)

Γ

is bounded with respect to the norm · L2 (Γ,H) . As (by definition) C 0 (Γ, H) is dense in L2 (Γ, H), this implies that IΓ admits a uniquely determined continuous linear extension to L2 (Γ, H), which we denote also by IΓ . We define f (z)dz := IΓ (f ) for f ∈ L2 (Γ, H) . Γ

Indeed, let |Γ| be the Euclidean length of Γ. Then, for each f ∈ L2 (Γ, H), we get from the Cauchy-Schwarz inequality: 1/2 1/2

IΓ (f ) ≤

f (z) H |dz| ≤ |dz|

f (z) 2H |dz| Γ

= |Γ| f L2 (Γ,H) .

Γ

Γ


279

8.3.5 Definition. Let f ∈ L2 (Γ, H), and let U ⊆ C ∪ {∞} be a neighborhood of Γ. (i) Let f − ∈ OH U ∩ (D− ∪ {∞}) . We shall say that f − is a holomorphic extension of f if there exist a neighborhood V ⊆ U of Γ and a sequence fn− ∈ OH (Γ) ∩ OH (V ∩ D− ), n ∈ N, such that lim f − fn− L2 (Γ,H) = 0

n→∞

(8.3.2)

and lim max f − (z) − fn− (z) H = 0 if K ⊆ V ∩ D− is compact .

n→∞ z∈K

(8.3.3)

(ii) Let f + ∈ OH (U ∩ D+ ). We shall say that f + is a holomorphic extension of f if there exists a neighborhood V ⊆ U of Γ and a sequence fn+ ∈ OH (Γ) ∩ OH (V ∩ D+ ), n ∈ N, such that lim f − fn+ L2 (Γ,H) = 0

n→∞

(8.3.4)

and lim max f + (z) − fn+ (z) H = 0

n→∞ z∈K

if K ⊆ V ∩ D+ is compact .

(8.3.5)

8.3.6 Proposition. Let f ∈ L2 (Γ, H), let U ⊆ C ∪ {∞} be a neighborhood of Γ, and let W be a neighborhood of Γ with piecewise C 1 -boundary ∂W , oriented by W , such that W ⊆ U ∩ C. (i) Assume f − ∈ OH U ∩ (D− ∪ {∞}) is a holomorphic extension of f . Then f − (z) dz = f (z) dz . (8.3.6) D− ∩ ∂W

Γ

Moreover, if w ∈ W ∩ D− , then the Cauchy formula holds: f − (z) f (z) 1 1 f − (w) = dz − dz . 2πi z−w 2πi z−w D− ∩ ∂W

Γ

(ii) Assume f + ∈ OH U ∩ D+ is a holomorphic extension of f . Then f + (z) dz = − f (z) dz . D+ ∩∂W

(8.3.8)

Γ

Moreover, if w ∈ W ∩ D+ , then the Cauchy formula holds: f + (z) f (z) 1 1 f + (w) = dz + dz . 2πi z−w 2πi z−w D+ ∩ ∂W

(8.3.7)

Γ

(8.3.9)

280


Proof. The proofs of parts (i) and (ii) are similar. We may restrict ourselves to part (i). By hypothesis, there exist a neighborhood V ⊆ U of Γ and a sequence fn− ∈ H O (Γ) ∩ OH (V ∩ D− ), n ∈ N, with (8.3.2) and (8.3.3). After shrinking V we may assume that V ⊆ W and the boundary ∂V of V is also piecewise of class C 1 . We orient ∂V by V . Then, by Cauchy’s theorem, f − (ζ) dζ = f − (ζ) dζ . (8.3.10) D− ∩∂W

D− ∩∂V

Moreover, by Cauchy’s theorem, fn− (ζ) dζ = fn− (ζ) dζ D− ∩∂V

for all n .

Γ

By (8.3.3) and (8.3.2), this implies f − (ζ) dζ = f − (ζ) dζ . D− ∩∂V

Γ

Together with (8.3.10) this proves (8.3.6). Now we consider some point w ∈ W ∩ D− . Choose a neighborhood W of Γ with piecewise C 1 -boundary ∂W , oriented by W , so small that W ⊆ W and w ∈ W \ W . Then, by Cauchy’s formula, f − (z) f − (z) 1 1 − f (w) = dz − dz . (8.3.11) 2πi z−w 2πi z−w D− ∩ ∂W

D− ∩ ∂W

Since f − is a holomorphic extension of f , it follows easily that the function W ∩ D− z → f − (z)/(z − w) is a holomorphic extension of the function Γ z → f (z)/(z − w) . Therefore, to the second integral in (8.3.11) we can apply (8.3.6) with W instead of W , f (z)/(z − w) instead of f (z) and f − (z)/(z − w) instead of f − (z). This gives (8.3.7). 8.3.7. Since holomorphic functions in a neighborhood of Γ are uniquely determined by their values on Γ, from (8.3.7) and (8.3.9) we get: If f ∈ L2 (Γ, H) admits a holomorphic extension to some open set of the form W ∩ D− or W ∩ D+ , where W is a neighborhood of Γ such that each connected component of W intersects Γ, then this extension is uniquely determined.


281

The opposite is also true: 8.3.8 Proposition. (i) Let f, g ∈ L2 (Γ, H), let U be a neighborhood of Γ, and let H h ∈ O U ∩ D− such that h is both a holomorphic extension of f and a holomorphic extension of g. Then f = g. (ii) Let f, g ∈ L2 (Γ, H), let U be a neighborhood of Γ, and let h ∈ OH U ∩ D+ such that h is both a holomorphic extension of f and a holomorphic extension of g. Then f = g. Proof. The proofs of (i) and (ii) are similar. We restrict ourselves to the proof of part (i). Set h = f − g. We have to prove that h = 0. By hypothesis, the zero function on D− is a holomorphic extension of h. This means, by definition, that there exist a neighborhood V of Γ and a sequence hn ∈ OH (Γ) ∩ OH (V ∩ D− ), n ∈ N, such that lim h − hn L2 (Γ,H) = 0

n→∞

(8.3.12)

and lim max hn (z) H = 0 if K ⊆ V ∩ D− is compact .

n→∞ z∈K

(8.3.13)

Let {ej }j∈I be an orthonormal basis of H, and let hj ∈ L2 (Γ, C) and hnj ∈ OC (Γ) be the functions with h= hj ej and hn = hnj ej . (8.3.14) j∈I

j∈I

For each j ∈ I, it follows from (8.3.12) and (8.3.13) that lim hj − hnj L2 (Γ,C) = 0

n→∞

and

lim max hnj (z) = 0 if K ⊆ V ∩ D− is compact.

n→∞ z∈K

If W ⊆ V is a further (arbitrarily small) neighborhood of Γ and ϕ ∈ OC (W ), then this yields, for all j ∈ I, lim hj ϕ − hnj ϕ L2 (Γ,C) = 0

n→∞

and

lim max hnj (z)ϕ(z) = 0 if K ⊆ W ∩ D− is compact.

n→∞ z∈K

(8.3.15)

(8.3.16)

Hence, for each ϕ ∈ OC (Γ) and j ∈ I, the (scalar) zero function on D− is a holomorphic extension of hj ϕ. By (8.3.6) this implies that hj (z) ϕ(z) dz = 0 for all ϕ ∈ OC (Γ) and j ∈ I . (8.3.17) Γ

282


Let θ : Γ → C be the function with dz = θ|dz|. As Γ is piecewise C 1 , this function is piecewise continuous and |θ| = 1. Then (8.3.17) takes the form 5 6 hj , ϕθ 2 =0 for all ϕ ∈ OC (Γ) and j ∈ I . (8.3.18) L (Γ,C)

Since OC (Γ) is dense in L2 (Γ, C), also the functions of the form ϕ θ, ϕ ∈ OC (Γ), are dense in L2 (Γ, C). Therefore (8.3.18) means that hj = 0 for all j ∈ I. By (8.3.14) this yields h = 0. 8.3.9 Proposition. Let f ∈ L2 (Γ, H), let U ⊆ C ∪ {∞} be a neighborhood of Γ, and let h ∈ OH (U \ Γ) such that both hU ∩(D− ∪{∞}) and hU ∩D+ are holomorphic extensions of f . Then h admits a holomorphic extension h to U and h = f . Γ

Proof. It is sufficient to prove that h admits a holomorphic extension h to U . The relation h Γ = f then follows from Proposition 8.3.8. Choose a neighborhood W of Γ with piecewise C 1 -boundary, oriented by W , such that W ⊆ U ∩ C. We define a holomorphic function F : W → C, setting 1 h(z) h(w) = dz , w∈W. 2πi z−w ∂W

We have to prove that h(w) = h(w) for w ∈ W \ Γ. The proofs are similar for w ∈ W ∩ D− and w ∈ W ∩ D+ . Let w ∈ D− ∩ W be given. Since hU ∩D+ is a holomorphic extension of f , it follows that the function D+ ∩ W z −→

1 h(z) 2πi z − w

(8.3.19)

is a holomorphic extension of the function Γ z −→

1 f (z) . 2πi z − w

(8.3.20)

Therefore, by (8.3.8) (with (8.3.19) instead of f + and (8.3.20) instead of f ), 1 h(z) f (z) 1 dz = − dz . 2πi z−w 2πi z−w D+ ∩ ∂W

Γ

Moreover, from the Cauchy formula (8.3.7) we get 1 h(z) f (z) 1 h(w) = dz − dζ . 2πi z−w 2πi z−w D− ∩ ∂W

Together this proves that h(w) = h(w).

Γ


283

8.3.10 Theorem and Definition. Recall that, by Theorem 3.7.3 and Proposition 3.1.3, the space OH (Γ) is the (algebraic) direct sum of OH (D+ ) and O0H (D− ∪ {∞}). We denote by P = PΓ the linear projector from OH (Γ) to OH (D+ ) parallel to O0H (D− ∪ {∞}), and we set Q = I − P. (i) The projector P is continuous with respect to the norm · L2 (Γ,H) . Since OH (Γ) is dense in L2 (Γ, H) (Section 8.3.3), it follows that P admits a uniquely determined continuous linear extension to L2 (Γ, H). This extension will be also denoted by P, and we set Q = I − P, L2+ (Γ, H) = PL2 (Γ, H) and L2− (Γ, H) = QL2 (Γ, H). − (ii) For each f ∈ L2 (Γ, H), there exist uniquely determined functions f ∈ O0H D− ∪ {∞} and f + ∈ OH D+ such that f + is the holomorphic extension of Pf 3 , and f − is the holomorphic extension of Qf . These functions are given by

1 f+ (z) = 2πi

Γ

1 f (ζ) dζ = ζ −z 2πi

(Pf )(ζ) dζ , ζ −z

Γ

z ∈ D+ ,

(8.3.21)

and f− (z) = −

1 2πi

Γ

f (ζ) 1 dζ = − ζ −z 2πi

Γ

(Qf )(ζ) dζ , ζ −z

z ∈ D− .

(8.3.22)

(iii) Let Γ be the unit circle, and let {ej }j∈N be an orthonormal basis of H. Then the functions zn √ ej , n ∈ Z, j ∈ N, (8.3.23) 2π form an orthonormal basis of L2 (Γ, C), and, hence, by Laurent decomposition, the family zn √ ej , n ∈ N, j ∈ N, (8.3.24) 2π forms an orthonormal basis of L2+ (Γ, C), and the family zn √ , 2π

n ∈ Z, n ≤ −1, j ∈ N,

(8.3.25)

forms an orthonormal basis of L2− (Γ, C). In particular, then P is an orthogonal projector. 3 By

the observation in Section 8.3.7, here we may speak about the holomorphic extension.

284


Proof. (iii) We have n 2 z √ |dz| = 1 2π 2π |z|=1

|dz| = 1

for all n ∈ Z ,

|z|=1

and

2π

znzm

|dz| =

|z|=1

z

n−m

|dz| =

|z|=1

=

ei(n−m)t dt 0

t=2π 1 =0 ei(n−m)t i(n − m) t=0

for all n, m ∈ Z with n = m .

If Γ is the unit circle, this shows that zn √ , 2π

n ∈ Z,

is an orthonormal system in L2 (Γ, C), which further implies that (8.3.23) is orthonormal in L2 (Γ, H). Moreover if Γ is the unit circle and f (z) =

∞

z n fn

n=−∞

is the Laurent expansion of a function f ∈ OH (Γ), then f (z) =

∞ ∞

fn , ej

H

z n ej .

n=−∞ j=0

This implies that f belongs to the closed linear hull of (8.3.23). As OH (Γ) is dense in L2 (Γ, H), this completes the proof of part (iii). (i) We first discuss the scalar case H = C, where the main difficulties already appear. In this discussion, we may restrict ourselves to the case when Γ is connected and, hence, D+ is simply connected. The general case then easily follows applying this special case to each of the connected components of Γ. If Γ is of class C 1 (and not only piecewise C 1 , as usually in this book), for a proof we can refer to the book [GKru]4 , where the proof is reduced to the case of the unit circle using a conformal mapping from D+ to the unit disc and its boundary properties. For the general case (when Γ is only piecewise C 1 ) we can refer only to the original paper [CMM], where the much more general case of Lipschitz contours is considered.5 4 In

[GKru] even the more general case of Ljapunov contours is considered. do not know whether there exists in the literature a more simple direct proof for the case of piecewise C 1 -contours. 5 We


285

We now give a proof of part (i), using the fact that this is already known for H = C. Let {ej }j∈I be an orthonormal basis of H. Denote by F the subspace of L2 (Γ, H) which consists of the functions of the form f=

ϕj (z)ej ,

z ∈ Γ,

j∈J

where J ⊆ I is finite and ϕj ∈ OC (Γ), j ∈ J. For such functions it is easy to see that 2 ϕj 2 , Pf = Pϕj ej

f 2L2 (Γ,H) = L (Γ,C) j∈J

and

Pf 2L2 (Γ,H) =

j∈J

2 2 Pϕj 2 2 ϕj 2 P 2 ≤ , L (Γ,C) L (Γ,C) L (Γ,C) j∈J

j∈J

where P L2 (Γ,C) denotes the norm of P as an operator in L2 (Γ, C). If P L2 (Γ,H) is the norm of P as an operator in L2 (Γ, H), this implies that

Pf L2 (Γ,H) ≤ P L2 (Γ,C) f L2 (Γ,H)

for all f ∈ F .

(8.3.26)

Set fj (z) = f (z), ej H Then

f 2L2 (Γ,H) =

for z ∈ Γ and f ∈ OH (Γ) .

2

fj L2 (Γ,C) .

j∈I

for each f ∈ OH (Γ). Therefore, for each f ∈ OH (Γ) and each ε > 0, there exists a finite set J ⊆ I with fj ej 2 < ε. f − j∈J

L (Γ,H)

Since OH (Γ) is dense in L2 (Γ, H), this implies that F is dense in L2 (Γ, H). Together with (8.3.26) this completes the proof of part (i). (ii) Let f ∈ L2 (Γ, H) be given. We define f + by the first equality in (8.3.21), and we define f − by the first equality in and (8.3.22). It is clear that in this way functions f + ∈ OH (D+ ) and f − ∈ O0H D− ∪ {∞} are well defined. It remains to prove the second equality in (8.3.21), the second equality in (8.3.22) and the facts that f + is a holomorphic extension of Pf and f − is a holomorphic extension of Qf . Since OH (Γ) is dense in L2 (Γ, H), we can choose a sequence hn ∈ OH (Γ), n ∈ N, with (8.3.27) lim hn − f L2 (Γ,H) = 0 . n→∞

286


Since, by part (i) of the theorem, P and Q are continuous, this implies that lim Phn − Pf L2 (Γ,H) = lim Qhn − Qf L2 (Γ,H) = 0 .

n→∞

n→∞

(8.3.28)

Moreover, by Cauchy’s formula and Cauchy’s theorem, 1 (Phn )(z) = 2πi

Γ

(Phn )(ζ) 1 dζ = ζ −z 2πi

Γ

hn (ζ) dζ ζ −z

(8.3.29)

for z ∈ D+ , and 1 (Qhn )(z) = − 2πi

Γ

(Qhn )(ζ) 1 dζ = − ζ −z 2πi

Γ

hn (ζ) dζ ζ −z

(8.3.30)

for z ∈ D− . The second equality in (8.3.21) follows passing to the limit in (8.3.29), and the second equality in (8.3.22) follows passing to the limit in (8.3.30). Now let K be a compact subset of D+ . Then, by (8.3.29) and the definition of f + ,

hn (ζ) − f (ζ) H 1 + max (Phn )(z) − f (z) H ≤ |dζ| . z∈K 2π |ζ − z| Γ

If |Γ| is the length of Γ and d is the distance between K and Γ, using the CauchySchwarz inequality, this further implies 1 max (Phn )(z) − f + (z)H ≤ |Γ|1/2 hn − f L2 (Γ,H) . z∈K 2πd Together with (8.3.28) this implies that lim max (Phn )(z) − f + (z)H = 0 .

n→∞ z∈K

Hence f + is a holomorphic extension of Pf . In the same way it follows from (8.3.30) and (8.3.28) that f − is a holomorphic extension of Qf . 8.3.11 Corollary (to the preceding theorem and definition). Since OH (Γ) is dense in L2 (Γ, H) and P and Q are continuous with respect to the topology of L2 (Γ, H), it follows immediately from the definitions of L2+ (Γ, H) and L2− (Γ, H) that the space OH (D+ ) is dense in L2+ (Γ, H) and the space O0H D− ∪ {∞} is dense in L2− (Γ, H).

8.4. Operator functions with values acting in a Hilbert space

8.4

287

Operator functions with values acting in a Hilbert space

In this section, H is a separable Hilbert space6 , D+ ⊆ C is a bounded connected open set with piecewise C 1 -boundary Γ such that 0 ∈ D+ , and D− := C \ D+ . Further, throughout this section L2 (Γ, H), L2+ (Γ, H) and L2− (Γ, H) denote the Hilbert spaces introduced in Section 8.3.1 and Theorem and Definition 8.3.10, and P denotes the projector from L2 (Γ, H) onto L2+ (Γ, H) parallel to L2− (Γ, H). 8.4.1 Definition. Let A : Γ → L(H) be a continuous function. Then we denote by Γ WA (or by WA ) the bounded linear operator acting in L2+ (Γ, H) by f ∈ L2+ (Γ, H) . WA f = P Af , This operator WA will be called the Wiener-Hopf operator defined by A on L2+ (Γ, H). Sometimes we use also the notation MA to denote the operator acting in L2 (Γ, H) by multiplication by A. Then WA = PMA L2 (Γ,H) . +

8.4.2 Theorem. Let A : Γ → GL(H) be a continuous function which admits local factorizations with respect to Γ and GL(H) (Def. 7.1.3), and let WA be the WienerHopf operator defined by A on L2+ (Γ, H). Then: (i) The function A admits a canonical factorization with respect to Γ and GL(H) (Def. 7.1.1), if and only if, WA is invertible. (ii) The function A admits a factorization with respect to Γ and GL(H) (Def. 7.1.1), if and only if, WA is a Fredholm operator. If this is the case, with the notations from Definition 7.1.1, dim Coker WA = κj dim Pj , 1≤j≤n , κj >0

dim Ker WA = −

(8.4.1) κj dim Pj ,

1≤j≤n , κj 0 resp. if there is no j with κj < 0. The remainder of this section is devoted to the proof of this theorem. We will deduce it from Theorem 8.1.4. The first step is the following lemma. 8.4.3 Lemma. Let W be a bounded neighborhood of Γ such that each connected component of W contains at least one connected component of Γ. We set W+ = 6 For

simplicity we consider only separable Hilbert spaces, although the results can be generalized to the non-separable case.

288


D+ ∪ W and W− = D− ∪ W . Let A : W → GL(H) be a continuous function which is holomorphic in W . Let WA be the Wiener-Hopf operator defined by A on 7A be the Wiener-Hopf operator defined by A on L2+ (Γ, H) (Def. 8.4.1), and let W H O (W + ) (Def. 8.1.2). Then

and

7A Ker WA = Ker W

(8.4.2)

7A . OH (W + ) ∩ Im WA = Im W

(8.4.3)

Proof. (See Def. 8.1.1 for the notations.) We first prove (8.4.2). The relation 7A Ker WA ⊇ Ker W is obvious. To prove the opposite relation, let f ∈ Ker WA be given. Then g := Af ∈ L2− (Γ, H). By statement (ii) in Theorem and Definition 8.3.10, g admits a holomorphic extension g− to D− . Since A−1 is holomorphic on W and continuous on W , this implies that f = A−1 g admits the holomorphic extension A−1 g− to W ∩ D− , which further extends continuously to W ∩ D− . On the other hand, by the same statement (ii) in Theorem and Definition 8.3.10, f admits a holomorphic extension to D+ . By Proposition 8.3.9, together this implies that f ∈ OH (W + ) .

(8.4.4)

Hence Af ∈ OH (W ). Since, moreover, Af ∈ L2− (Γ, H), it follows that Af ∈ OH 0 (W − ∪ {∞}). Together with (8.4.4) this means that f ∈ Ker W A . Now we prove (8.4.3). The relation OH (W + ) ∩ Im WA ⊇ Im W A is obvious. To prove the opposite relation, let f+ ∈ OH (W + ) ∩ Im WA be given. Then there exist u+ ∈ L2+ (Γ, H) and g− ∈ L2− (Γ, H) such that f+ + g− = Au+ .

(8.4.5)

Hence g− = Au+ − f+ . Since A is continuous W and holomorphic in W , by on Proposition 8.3.9 this implies that g− ∈ OH W and hence (as g− ∈ L2− (Γ, H)) (8.4.6) g− ∈ OH 0 W − ∪ {∞} . Since f+ ∈ OH W + , in view of (8.4.5) this further implies that Au+ ∈ OH W . As A−1 is continuous on W and holomorphic in W , so we get u+ ∈ OH W and hence (as u+ ∈ L2+ (Γ, H)) (8.4.7) u+ ∈ O H W + . 7A . From (8.4.5)–(8.4.7) it follows that f ∈ Im W

8.4. Operator functions with values acting in a Hilbert space

289

8.4.4 Proposition. (see the beginning of Section 8.3 for the notations) The space OH (D+ ) is contained in L2+ (Γ, H) as a dense subspace, and OH D ∪ {∞} is − 0 contained in L2− (Γ, H) as a dense subspace. H Proof. By the Mergelyan approximation Theorem 2.2.1, O (D+ ) is dense in H H H O (D+ ) and O D− ∪ {∞} is dense in O D− ∪ {∞} with respect to uniform convergence and, hence, with respect to the topology of L2 (Γ, H). Since, by defini tion of P and Q, Pf = f for f ∈ OH (D+ ) and Qf = f for f ∈ OH D− ∪{∞} and since P and Q are continuous on L2 (Γ, H), this implies that OH (D+ ) is contained 2 H in L+ (Γ, H) and O D− ∪ {∞} is contained in L2− (Γ, H). To prove the density, let f + ∈ L2+ (Γ, H) and f − ∈ L2− (Γ, H) be given. Since H O (Γ) is dense in L2 (Γ, H) (cf. Section 8.3.3), then there are sequences h+ n and H h− in O (Γ) such that n + lim h+ n − f L2 (Γ,H) = 0 and

n→∞

− lim h− n − f L2 (Γ,H) = 0

n→∞

and hence, as P and O are continuous, + lim Ph+ n − f L2 (Γ,H) = 0 and

n→∞

− lim Qh− n − f L2 (Γ,H) = 0.

n→∞

H − H Since Ph+ n ∈ O (D + ) and Qhn ∈ O (D − ∪ {∞}), this completes the proof of the density.

8.4.5 Lemma. Let A, B : Γ → L(E) be continuous functions, and let T− : D− ∪ {∞} → GL(E) and T+ : D+ → GL(E) be continuous functions, which are holomorphic in D− ∪ {∞} and D+ , respectively, such that A = T− BT+

on Γ.

(8.4.8)

Then: (i) WA = WT WB WT . −

+

(ii) If T− (z) is invertible for all z ∈ D− ∪ {∞}, then WT− is invertible and WT−1 = WT −1 . −

−

(iii) If T+ (λ) is invertible for all z ∈ D+ , then WT+ is invertible and WT−1 = WT −1 . +

+

In particular, if A and B are equivalent relative to Γ and GL(E) (Def. 7.1.3), then WA and WB are equivalent.7 7 Two

operators T and S in a Banach space X are called equivalent if there exist invertible operators V, W in X such that T = V SW .

290


Proof. It is sufficient to prove part (i), because (ii) and (iii) then follow. It follows from Proposition 8.4.4 that MT L2+ (Γ, H) ⊆ L2+ (Γ, H) and +

MT L2− (Γ, H) ⊆ L2− (Γ, H). Therefore −

PMT+ P = MT+ P

(8.4.9)

and PMT− (I − P) = 0 . From the second relation it follows that PMT− P = PMT− . From (8.4.8)–(8.4.10) together we obtain the assertion: WT− WB WT+ = PMT− PMB PMT+ 2 L+ (Γ,H) = PMT− MB MT+ 2 = PMA L+ (Γ,H)

(8.4.10)

L2+ (Γ,H)

= WA .

Proof of Theorem 8.4.2. First assume that A admits a factorization A = A− ΔA+ with respect to Γ and GL(H). Using Proposition 8.3.9, then it is easy to prove that WΔ is a Fredholm operator, where, with the notations from Definition 7.1.1, dim Coker WΔ = κj dim Pj , 1≤j≤n , κj >0

dim Ker WΔ = −

(8.4.11) κj dim Pj .

1≤j≤n , κj 0 such that the following holds: Any continuous operator function A : Γ → GL(H), which admits local factorizations with respect to Γ (Def. 7.1.3) and which satisfies max A(z) − I L(H) < ε , z∈Γ

(8.5.1)

admits a canonical factorization with respect to Γ (Def. 7.1.1). Proof. Let L2 (Γ, H), L2+ (Γ, H) and L2− (Γ, H) be the Hilbert spaces introduced in Section 8.3.1 and Theorem and Definition 8.3.10, and let P be the projector from L2 (Γ, H) onto L2+ (Γ, H) parallel to L2− (Γ, H). Recall that (by Theorem and Definition 8.3.10) P is a bounded linear operator on L2− (Γ, H), and set ε=

1

P L(L2 (Γ,H))

.

Now let a continuous function A : Γ → GL(H) be given which satisfies (8.5.1), and let MA−I : L2 (Γ, H) → L2 (Γ, H) be the operator of multiplication by A − I, and let WA be the Wiener-Hopf operator defined by A on L2+ (Γ, H) (Def. 8.4.1). Then 1 .

MA−I L(L2 (Γ,H)) < ε =

P L(L2 (Γ,H)) Since WA − I = PMA−I L2 (Γ,H) , this implies +

WA − I L(L2 (Γ,H)) ≤ P L(L2 (Γ,H)) MA−I L(L2 (Γ,H)) < 1 . +

Hence WA is invertible. If, in addition, A admits local factorizations with respect to Γ and GL(H), it follows from Theorem 8.4.2 that A admits a canonical factorization with respect to Γ. 8.5.2 Corollary (to Theorem 8.5.1). Let 0 < α < 1 and k ∈ N, where, for k ≥ 1, we additionally assume that Γ is of class C k (Def. 3.4.1). Then there exists ε > 0 such that, for any Ck+α -function A : Γ → GL(H) (Def. 3.4.3), which satisfies max A(z) − I L(H) < ε , z∈Γ

the following holds:

292


(i) A admits a canonical factorization with respect to Γ. (ii) If A = A− A+ is an arbitrary canonical factorization of A with respect to Γ, then automatically, the factors A− and A+ are of class C k+α on D− and D+ , respectively. Proof. Part (i) of Theorem 7.3.1 in particular yields that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 8.5.1. Part (ii) of the corollary follows from part (ii) of Theorem 7.3.1 8.5.3 Theorem. Let Γ = T be the unit circle. Then any continuous function A : T → GL(H), which admits local factorizations with respect to T (Def. 7.1.3) and which satisfies max A(z) − I L(H) < 1 , (8.5.2) z∈Γ

admits a canonical factorization with respect to T (Def. 7.1.1). Proof. This is a repetition of the proof of Theorem 8.5.1, taking into account that now (as Γ = T), by statement (iii) in definition and Theorem 8.3.10,

P L(L2 (Γ,H)) = 1 . 8.5.4 Corollary (to Theorem 8.5.3). Let Γ = T be the unit circle, and let A : Γ → GL(H) be a Ck+α -function, 0 < α < 1, k ∈ N (Def. 3.4.3), which satisfies max A(z) − I L(H) < 1 . z∈T

Then: (i) A admits a canonical factorization with respect to T. (ii) If A = A− A+ is an arbitrary canonical factorization of A with respect to T, then automatically, the factors A− and A+ are of class Ck+α on D− and D+ , respectively. Proof. Part (i) of Theorem 7.3.1 in particular yields that A admits local factorizations with respect to T. Therefore part (i) of the corollary follows from Theorem 8.5.3. Part (ii) of the corollary follows from part (ii) of Theorem 7.3.1 8.5.5 Corollary (to Theorem 8.5.3). Let Γ = T be the unit circle, and let R be a Banach algebra of continuous L(H)-valued functions satisfying conditions (A), (B) and (C) in Section 7.2.4. For example, let R = W L(H) be the Wiener algebra (see Section 7.2.1). Let A : T → GL(H) be a function which belongs to R and satisfies max A(z) − I L(H) < 1 . z∈T

Then:

8.5. Functions close to the unit operator or with positive real part

293

(i) A admits a canonical factorization with respect to T. (ii) If A = A− A+ is an arbitrary canonical factorization of A with respect to Γ, then automatically, the factors A− and A+ belong to the algebra R. Proof of Theorem 8.5.1. Part (ii) of Theorem 7.2.5 yields that A admits local factorizations with respect to Γ. Therefore part (i) of the corollary follows from Theorem 8.5.3. Part (ii) of the corollary follows from part (iii) of Theorem 7.2.5. 8.5.6. It is clear that the assertion of Theorem 8.5.3 remains valid if we replace the unit circle T by an arbitrary circle in C. However this is not true for more general contours. V.I. Macaev and A.I. Virozub [ViMa] even proved the following: If the assertion of Theorem 8.5.3 is valid for any finite dimensional Hilbert space H with T replaced by an arbitrary Jordan curve, then this Jordan curve is a circle. However there is the following result for operator functions of a special form. 8.5.7 Theorem. Assume that the contour Γ (of the generality as described at the beginning of this section) is connected, i.e., we assume that D+ is simply connected (Section 2.5.1). Let A : Γ → GL(H) be a function of the form A(z) =

T + B+ (z) , z − z0

z ∈ Γ,

(8.5.3)

where z0 ∈ D+ , T ∈ L(H) and B+ is a continuous L(H)-valued function on D+ , which is holomorphic in D+ , and assume that max A(z) − I L(H) < 1 .

(8.5.4)

z∈Γ

Then A admits a canonical factorization with respect to Γ and GL(H). 8.5.8. Of course, also for this theorem there is a corollary for functions of class C k+α corresponding to Corollary 8.5.2 of Theorem 8.5.1. Proof of Theorem 8.5.7. As A is of the form (8.5.3), it is continuous on D+ \ {z0 } and holomorphic in D+ \ {z0 }. Choose a neighborhood U ⊆ C \ {z0 } of Γ so small that q := sup A(z) − I L(H) < 1 . (8.5.5) z∈U ∩D +

Set

T = z ∈ C |z| = 1

and

D = z ∈ C |z| < 1 .

Since D+ is simply connected, we can find a biholomorphic map ϕ from a neighborhood of D onto D+ such that Γ := ϕ(T) ⊆ U ,

z0 ∈ D+ := ϕ(D)

and

ϕ(0) = z0 .

294


It is easy to see that then any canonical factorization of A with respect to Γ and GL(H) is also a canonical factorization with respect to Γ and GL(H). Therefore it is sufficient to prove that A admits a canonical factorization with respect to Γ and GL(H). For that we want to apply theroem 8.4.2. Since A is holomorphic in a neighborhood of Γ , it is trivial that A admits local factorizations with respect to Γ and GL(H). Therefore, to apply this theroem, we only have to prove that the Wiener-Hopf operator defined by A in L2+ (Γ , H) is invertible. Since the derivative of a biholomorphic map does not vanish, ρ(z) :=

|z − z0 |2 , |ϕ (ϕ−1 (z))|

z ∈ Γ ,

is a well-defined continuous function on Γ . Moreover, h(z) = 0 for all z ∈ Γ (as z0 ∈ Γ ). Therefore, setting

f, g ρ = f (z), g(z) H ρ(z) |dz| , Γ

we get a scalar product ·, ·ρ on L2 (Γ , H), such that the corresponding norm · ρ is equivalent to the norm · L2 (Γ ,H) introduced in Definition 8.3.1. Therefore, to prove the invertibility of the above mentioned Wiener-Hopf operator, now it is sufficient to show that

P(Af ) − f ρ ≤ q f ρ

(8.5.6)

for all f ∈ L2+ (Γ , H), where P is the projector from L2 (Γ , H) onto L2+ (Γ , H) parallel to L2− (Γ , H). Since OH (D+ ) is dense in L2+ (Γ , H) (Corollary 8.3.11), it is sufficient to prove this for all f ∈ OH (D+ ). Let such f be given. Let T be the operator from (8.5.3), set v = T f (z0 ) Then

P(Af ) − f, h ρ = 8

=

and 8

= T 5

= T

v , z − z0

P(Af ) − f (z) ,

Γ

P(Af ) − f

T

8

h(z) =

P(Af ) − f

P(Af ) − f

ϕ(z) , ϕ(z) ,

v ϕ(z) − z0 v ϕ(z) − z0

ϕ(z) , v

6 H

v z − z0

9 H

9

z ∈ C \ D+ .

9 H

|z − z0 |2 |dz| |ϕ (ϕ−1 (z))|

|ϕ(z) − z0 |2 ϕ ϕ−1 ϕ(z) |ϕ (z)||dz| ϕ(z) − z0 |2 |dz|

H

ϕ(z) − z0 |dz| .

8.5. Functions close to the unit operator or with positive real part

295

Since f is holomorphic in a neighborhood of D+ and A is holomorphic in a neighborhood of Γ , the function under the last integral is holomorphic in a neighborhood of D. Moreover, as ϕ(0) = z0 , this function vanishes at zero. Therefore the Taylor expansion at zero of this function is of the form 5

P(Af ) − f

∞ 6 ϕ(z) , v an z n . ϕ(z) − z0 = H

n=1

Since this series converges uniformly on T and

2π z |dz| = n

T

eint dt =

1 int t=2π =0 e in t=0

for n ≥ 1 ,

0

it follows that P(Af ) − f, h ρ = 0 . Hence

P(Af ) − f ρ ≤ P(Af ) − f + h ρ . Since, by (8.5.3), P(Af ) + h = Af and therefore P(Af ) − f + h = Af − f , it follows that

P(Af ) − f ρ ≤ Af − f ρ . As, by (8.5.5), Af − f ρ ≤ q f ρ , this proves (8.5.6).

8.5.9 Theorem. Let Γ = T be the unit circle. Assume A : T → GL(H) is a continuous function which admits local factorizations with respect to T (Def. 7.1.3, such that, for some c > 0,

Re A(z)v, v H ≥ c v H for all v ∈ H and z ∈ T , (8.5.7) where ·, ·H denotes the scalar product of H. Then A admits a canonical factorization with respect to T (Def. 7.1.1). Proof. By Theorem 8.4.2 it is sufficient to prove that the Wiener-Hopf operator WA defined by A on L2 (T, H) is invertible. It follows from (8.5.7) that, for each continuous function f : T → H,

Re Af, f L2 (T,H) = Re A(z)f (z), f (z) H |dz| T

≥c

f (z) H |dz| = cf 2L2 (T,H) .

T

Since the continuous functions are dense in L2 (T, H), this implies that

Re Af, f L2 (T,H) ≥ cf 2L2 (T,H) for all f ∈ L2 (T, H) .

296


Since, by statement (iii) in Theorem and Definition 8.3.10, the projector P is orthogonal and Pf = f for f ∈ L2+ (T, H), this implies that

Re WA f, f L2 (T,H) = Re P(Af ), f L2 (T,H) = Re Af, f L2 (T,H) 2 ≥ cf L2 (T,H) for all f ∈ L2+ (T, H) . Hence the real part of WA is positive, which implies that WA is invertible.

8.5.10 Theorem. Let Γ = T be the unit circle. Assume A : Γ → GL(H) is a continuous function which admits local factorizations with respect to Γ (Def. 7.1.3), such that at least one of the following two conditions is satisfied: max A−1 (z) − I L(H) < 1 ,

(8.5.8)

z∈T

or, for some c > 0, ReA−1 (z)v, vH ≥ c v H

for all v ∈ H and z ∈ T .

(8.5.9)

Then A admits a canonical factorization with respect to T (Def. 7.1.1). Proof. Set B(z) = A−1

1 . z

Since A admits local factorizations with respect to T, then B admits local factorizations with respect to T. Moreover, since A satisfies at least one of the conditions (8.5.8) or (8.5.9), B satisfies at least one of the conditions max B(z) − I L(H) < 1 , z∈T

or ReB(z)v, vH ≥ c v H

for all v ∈ H .

Therefore, by theorems 8.5.3 and 8.5.9, B admits a canonical factorization B = B− B+ with respect to T. Setting −1 A− (z) = B+

1 z

and

−1 A+ (z) = B−

1 , z

we obtain a required canonical factorization A = A− A+ of A with respect to T. 8.5.11. Finally we note that also theorems 8.5.9 and 8.5.10 have corollaries corresponding to corollaries 8.5.4 and 8.5.4 of Theorem 8.5.3.

8.6. Block Töplitz operators

8.6

297

Block Töplitz operators

In this section H is a separable Hilbert space and l2 (H) = H ⊕ H ⊕ H ⊕ . . . is defined to be the Hilbert space of square integrable sequences v = (vn )n∈N of vectors vn ∈ H, endowed with the scalar product v, w =

∞

vn , wn ,

v = (vn )n∈N ∈ l2 (H),

w = (wn )n∈N ∈ l2 (H),

n=0

and the norm : ;∞ ;

vn 2 ,

v =
0.

(8.6.14)

On the other hand, as A(z0 ) is not left invertible, we can find v0 ∈ H with

v0 = 1

and

A(z0 ) v0 < α . 4

Choose δ > 0 so small that

δ < 2

|dz| < 2δ z∈T,|z−z0 | 1, but

Af L2 (T,H)
1.

n→∞

n→∞

(8.6.18)

302


On the other hand, by (8.6.17),

lim WA Pk (z n f )

L2 (T,H)

n→∞

= lim WA (z n f ) n→∞ L2 (T,H)

= lim P Az n f 2 . L (T,H)

n→∞

As the projector P is orthogonal and in view of (8.6.16), this further implies that

lim WA Pk (z n f ) 2 ≤ lim Az n f 2 n→∞ n→∞ L (T,H) L (T,H) α = Af < . 2 L2 (T,H) Together with (8.6.18) this is a contradiction to (8.6.14). Now we assume that A∗ (z0 ) is not left invertible. Since WA∗ is the Hilbert space adjoint of WA , which is also a Fredholm operator (as WA is a Fredholm operator), then, with A replaced by A∗ , we get the same contradiction. We now can prove the following two theorems: 8.6.7 Theorem. Let A : T → GL(H) be a continuous function which admits local factorizations relative to T and GL(H) (Def. 7.1.3). Then the following two conditions are equivalent: (i) The T¨ oplitz operator TA defined by A is invertible. (ii) A admits a canonical factorization relative to T and GL(H) (Def. 7.1.1). If these two equivalent conditions are satisfied and A = A− A+ is a canonical factorization of A relative to T and GL(H), then the inverse of TA is given by TA−1 = TA−1 TA−1 . +

(8.6.19)

+

Moreover, the block matrix of the operator TA−1 has the lower triangular form +

⎛

Γ+ 0 ⎜ Γ+ ⎜ 1+ ⎜Γ ⎝ 2 .. .

0 Γ+ 0 Γ+ 1 .. .

0 0 Γ+ 0 .. .

⎞

... . . .⎟ ⎟ . . .⎟ ⎠ .. .

Γ+ n :=

with

1 2πi

T

A−1 + (z) dz , z n+1

n ∈ N, (8.6.20)

the block matrix of the operator TA−1 has the upper triangular form −

⎛ − Γ0 ⎜ 0 ⎜ ⎜ 0 ⎝ .. .

Γ− −1 Γ− 0 0 .. .

Γ− −2 Γ− −1 Γ− 0 .. .

⎞

... . . .⎟ ⎟ . . .⎟ ⎠ .. .

with

Γ− −n

1 := 2πi

T

A−1 − (z) dz , z −n+1

n ∈ N, (8.6.21)

8.6. Block Töplitz operators

303

∞ and, hence, the block matrix Γjk j,k=0 of TA−1 can be computed by the finite sums k − Γ+ j−ν Γν−k , := ν=0 j + − ν=0 Γj−ν Γν−k ,

Γjk

j ≥ k, j ≤ k.

(8.6.22)

8.6.8 Theorem. Let A : T → GL(H) be a continuous function which admits local factorizations relative to T and GL(H) (Def. 7.1.3). Then the following two conditions are equivalent: (i) The T¨ oplitz operator TA defined by A is a Fredholm operator. (ii) A admits a factorization relative to T and GL(H) (Def. 7.1.1). If these two equivalent conditions are satisfied, if n κj z Pj A+ (z), A(z) = A− (z) P0 +

z ∈ T,

j=1

is a factorization of A relative to T and GL(H), and r is the index with κ1 > . . . > κr > 0 > κr+1 > . . . > κn , then dim Ker TA = −

n

κj dim Pj

and

dim Coker TA =

j=r+1

r

κj dim Pj . (8.6.23)

j=1

Moreover, if Δ(z) := P0 +

n j=1

z κj Pj , then TA−1 TΔ−1 TA−1 +

(8.6.24)

−

is a generalized inverse (Section 6.10.2) of TA , where, with the notations introduced in Section 8.6.4, TΔ−1 = S0,P0 +

n

S−κj ,Pj = S0,P0 +

j=1

r

κ

j S−1, + j

j=1

n

−κ

S1,Pjj .

(8.6.25)

j=r+1

8.6.9. Proof of Theorem 8.6.7. By (8.6.6) it is clear that the statement on the equivalence of conditions (i) and (ii) coincides with part (i) of Theorem 8.3.7. Now let A = A− A+ be a canonical factorization of A relative to T and GL(H). Then, by parts (ii) and (iii) of Proposition 8.6.5, the operators TA and +

TA are invertible, where TA−1 = TA−1 and TA−1 = TA−1 . Since, by part (i) of this −

+

−

+

proposition, TA = TA TA , this further implies that −

−

+

TA−1 = TA−1 TA−1 = TA−1 TA−1 . +

−

+

−

304


Moreover, by (8.6.6), the block matrix of the operator TA−1 is given by ⎛ + Γ0 ⎜ Γ+ ⎜ 1+ ⎜Γ ⎝ 2 .. .

Γ+ −1 Γ+ 0 Γ+ 1 .. .

+

⎞ ... . . .⎟ ⎟ . . .⎟ ⎠ .. .

Γ+ −2 Γ+ −1 Γ+ 0 .. .

with Γ+ n :=

1 2πi

T

A−1 + (z) dz , z n+1

n ∈ Z,

A−1 − (z) dz , z n+1

n ∈ Z.

and the block matrix of the operator TA−1 is given by ⎛

Γ− 0 ⎜ Γ− ⎜ 1− ⎜Γ ⎝ 2 .. .

Γ− −1 Γ− 0 Γ− 1 .. .

Γ− −2 Γ− −1 Γ− 0 .. .

⎞ ... . . .⎟ ⎟ . . .⎟ ⎠ .. .

Γ+ n =

1 2πi

As

and Γ− n

1 = 2πi

−

with Γ− n :=

T

T

1 2πi

T

A−1 + (z) dz = 0 z n+1

if n < 0,

A−1 − (z) dz = 0 z n+1

if n > 0,

this implies (8.6.20) and (8.6.21).

8.6.10. Proof of Theorem 8.6.8. Again in view of (8.6.6), the statement on the equivalence of conditions (i) and (ii) coincides with part (ii) of Theorem 8.3.7. Now let these two equivalent conditions be satisfied, let n A(z) = A− (z) P0 + z κj Pj A+ (z), z ∈ T, j=1

be a factorization of A relative to T and GL(H), and let r be the index with κ1 > . . . > κr > 0 > κr+1 > . . . > κn . Then (8.6.23) follows from part (ii) of Theorem 8.3.7. From part (i) of proposition (8.6.5) it follows that T A = T A− T Δ T A + .

(8.6.26)

By definition (see Section 8.6.4), we have TΔ = S0,P0 +

n

Sκj ,Pj

and TΔ−1 = S0,P0 +

j=1

n

S−κj ,Pj .

(8.6.27)

j=1

By (8.6.13) and (8.6.12) this yields TΔ TΔ−1 TΔ = S0,P0 +

n j=1

Sκj ,Pj S−κj ,Pj Sκj ,Pj = S0,P0 +

n j=1

Sκj ,Pj = TΔ .

8.7. The Fourier transform of L1 (R, E)

305

In the same way we get the relation TΔ−1 TΔ TΔ−1 = TΔ−1 . Hence TΔ−1 is a generalized inverse of TΔ . Taking into account (8.6.26) and the fact that, by parts (ii) and (iii) of proposition (8.6.5), TA−1 = TA−1 and TA−1 = TA−1 , +

+

−

−

this implies that the operator (8.6.24) is a generalized inverse of TA . The first equality in (8.6.25) holds by (8.6.27) and the second follows from (8.6.12).

8.7

The Fourier transform of L1 (R, E)

8.7.1. The space Lp (R, E). Let E be a Banach space, and let 1 ≤ p < ∞. We denote by C00 (R, E) the complex linear space of continuous functions f : R → E with compact support. We introduce a norm · Lp (R,E) on C00 (R, E), setting ∞

f (x) p dx, f ∈ C00 (R, E),

f pLp (R,A) = −∞

and we denote by Lp (R, E) the completion of C00 (R, E) with respect to this norm. For E = C this is the usual space of scalar Lp -functions on R. In simple cases, also for general Banach spaces E, we identify the elements of Lp (R, E) with Evalued functions. For example, each piecewise continuous function f : R → E ∞ with −∞ F (x) p dx < ∞ will be viewed (in the obvious way) as an element of Lp (R, E). Note that then ∞

f (x) p dx.

f pLp (R,A) = −∞

Also the functions of the form ϕ·v, where v is a fixed vector in E and ϕ ∈ Lp (R, C), will be identified with the corresponding element in Lp (R, E). In general however, we view the elements of Lp (R, E) as equivalence classes of Cauchy sequences rather than true functions (although the latter is also possible). Nevertheless, for each element f ∈ L1 (R, E), we define the integral ∞ f (x) dx (8.7.1) −∞

as follows: Take a Cauchy sequence fn ∈ C00 (R, E) defining f . Then ∞ fn (x) dx lim n→∞

−∞

exists and is independent of the choice of the Cauchy sequence. We define the integral (8.7.1) to be this limit.

306


8.7.2. Convolution. Let A be a Banach algebra. For two functions K, L ∈ L1 R, A , we introduce a product K ∗ L ∈ L1 (R, A), called the convolution of K and L. If K and L are piecewise continuous, this is defined (as usual) by (K ∗ L)(y) :=

∞

−∞

K(x)L(y − x)dx =

∞

−∞

K(y − x)L(x)dx,

y ∈ R.

By Fubini’s theorem, for such functions we have ∞ ∞

K ∗ L L1 (R,A) = K(y − x)L(x)dx dy −∞ −∞ ∞ ∞

K(y − x) L(x) dx dy ≤ −∞ −∞ ∞ ∞ =

K(y − x) dy L(x) dx −∞ −∞ ∞ ∞ =

K(y) dy L(x) dx = K L1 (R,A) L L1 (R,A) . −∞

−∞

Therefore, we can extend the convolution by continuity to all of L1 R, A , and then

K ∗ L L1 (R,A) ≤ K L1 (R,A) L L1 (R,A) (8.7.2) for all K, L ∈ L1 R, A . Hence L1 R, A is an algebra with respect to convolution. 8.7.3. spaces Lp± (R, E). Let E be a Banach space, let 1 ≤ p < ∞, and 0The let C0 − (R, E) be the subspace of C00 (R, E) which consists of the functions with support in ]−∞, 0], and let C00 + (R, E) be the subspace of C00 (R, E) which consists of the functions with support in [0, ∞[. Then we denote by Lp− (R, E) and Lp+ (R, E) the closures of C00 − (R, E) and 0 C0 + (R, E) in Lp (R, E), respectively. Since Lp− (R, E) ∩ L1+ (R, E) = {0} and the functions of the form f+ + f− with f± ∈ C00 ± (R, E) are dense in C00 (R, E) with respect to the norm · Lp (R,E) , we see that Lp (R, E) splits into the direct sum Lp (R, E) = Lp− (R, E) ⊕ Lp+ (R, E).

(8.7.3)

If A is a Banach algebra, then, obviously, f ∗ g ∈ L1+ (R, A)

if f, g ∈ L1+ (R, A),

f ∗ g ∈ L1− (R, A)

if f, g ∈ L1− (R, A),

(8.7.4)

i.e., L1+ (R, A) and L1− (R, A) are subalgebras of the algebra L1 (R, A) with respect to covolution.


307

8.7.4. The Fourier transform of L1 (R, E). Let E be a Banach space. For functions f ∈ C00 (R, E), we define by ∞ eiλx f (x) dx , λ ∈ C, (8.7.5) f!(λ) = −∞

the Fourier transform of f . Differentiating under the sign of integration, we see that f! is holomorphic on C for all f ∈ C00 (R, E). Moreover, if f ∈ C00 (R, E) and λ ∈ R, then ∞

f!(λ) ≤ |eiλx | f (x) dx = f 1 . −∞

L (R,E)

Therefore, the Fourier transformation extends by continuity to all of L1 (R, E), where, for all elements f ∈ L1 (R, E), f! is a continuous E-valued function on R, and sup f!(λ) ≤ f L1 (R,E) for all f ∈ f ∈ L1 (R, E). (8.7.6) λ∈R

We denote by L!1 (R, E), L!1+ (R, E) and L!1− (R, E) the spaces of all continuous functions ϕ : R → E of the form ϕ = f! with f ∈ L1 (R, E), f ∈ L1+ (R, E) and f ∈ L1− (R, E), respectively. Note that f! ≡ 0

if f ∈ L1 (R, E) and f ≡ 0.

(8.7.7)

This can be deduced by the Hahn-Banach theorem from the scalar case. Therefore the Fourier transform is a linear isomorphism from L1 (R, E) onto L!1 (R, E), and we can introduce a norm in L!1 (R, E), setting

f! L!1 (R,E) = f L1 (R,E)

for f ∈ L1 (R, E).

(8.7.8)

for all ϕ ∈ L!1 (R, E).

(8.7.9)

Note that then (8.7.6) takes the form sup ϕ(λ) ≤ ϕ L!1 (R,E)

λ∈R

As L1 (R, E) = L1+ (R, E)⊕L1− (R, E), the space L1 (R, E) splits into the direct sum L!1 (R, E) = L!1+ (R, E) ⊕ L1− (R, E). 8.7.5. Now we consider again a Banach algebra A. Then for K, L ∈ C00 R, A , it follows from Fubini’s theorem that ∞ ∞ eiλy K(y − x)L(x)dx dy (K ∗ L)(λ) = −∞ −∞ ∞ ∞ = eiλ(y−x) K(y − x) dy eiλx L(x)dx −∞ −∞ ∞ ∞ ! L(λ) ! = eiλ(y) K(y) dy eiλx L(x)dx = K(λ) −∞

−∞

(8.7.10)

(8.7.11)

308


for all λ ∈ C. Since the convolution is continuous with respect to the L1 -norm (inequality (8.7.2)) and in view of estimate (8.7.6), this implies that ! L(λ) ! (K ∗ L)(λ) = K(λ)

(8.7.12)

for all K, L ∈ L1 R, A and λ ∈ R. Hence, for all K, L ∈ L!1 R, A , the pointwise defined product AL belongs again to L!1 (R, A), and it follows from (8.7.2) and the definition of the norm in L!1 (R, A) that

KL L!1 (R,A) ≤ K L!1 (R,A) L L!1 (R,A)

(8.7.13)

for all K, L ∈ L!1 R, A . Hence L!1 R, A is a Banach algebra, and, as L1+ R, A and L1− R, A are subalgebras of L1 R, A with respect to convolution, L!1+ R, A and L!1− R, A are subalgebras of L!1 R, A . 8.7.6. We denote by H+ the upper open half plane, and by H− the lower open half plane: H− := z ∈ C Im z < 0 . H+ := z ∈ C Im z > 0 , The closures of H+ and H− in C (and not in the Riemann sphere) will be denoted by H+ and H− . If E is a Banach space, then a function f : H+ ∪ {∞} → E will be called continuous on H+ ∪ {∞} if f is continuous on H+ and f (1/z) is also continuous on H+ . In the same way we define what it means that a function defined on H− ∪ {∞} or on R ∪ {∞} is continuous. 8.7.7 Theorem. Let E be a Banach space. (i) A function ϕ ∈ L!1 (R, E) belongs to L!1+ (R, E), if and only if it admits a continuous extension to H+ ∪ {∞} which is holomorphic in H+ and vanishes at infinity. Moreover max

λ∈H+ ∪{∞}

ϕ(λ) ≤ ϕ L!1 (R,E)

for each ϕ ∈ L!1+ (R, E).

(ii) A function ϕ ∈ L!1 (R, E) belongs to L!1− (R, E), if and only if it admits a continuous extension to H− ∪ {∞} which is holomorphic in H− and vanishes at infinity. Moreover max

λ∈H− ∪{∞}

ϕ(λ) ≤ ϕ L!1 (R,E)

for each ϕ ∈ L!1− (R, E).

(8.7.14)


309

(iii) Each function ϕ ∈ L!1 (R, E) is continuous on R ∪ {∞} and ϕ(∞) = 0. Moreover max λ∈R∪{∞}

ϕ(λ) ≤ ϕ L!1 (R,E)

for each ϕ ∈ L!1 (R, E).

Proof. As L1 (R, E) = L1− (R, E) ⊕ L1+ (R, E), part (iii) follows from parts (i) and (ii) and estimate (8.7.6). By the same reason, for the proof of (i) and (ii), it is sufficient to prove the “only if” parts of (i) and (ii). Indeed, assume that this is proved. Let ϕ+ ∈ L!1 (R, E) be a function which admits a continuous extension to H+ ∪ {∞}, which is holomorphic in H+ and vanishes at infinity. Then it follows from the decomposition L1 (R, E) = L1− (R, E) ⊕ L1+ (R, E) that ϕ+ = ϕ − + ϕ +

on R ∪ {∞}

with ϕ ± ∈ L!1± (R, E). Then ϕ+ − ϕ + = ϕ −

on R ∪ {∞}.

The two sides of this equation define a holomorphic function on C∪{∞} vanishing at infinity (see Theorem 1.5.4). By Liouville’s theorem this implies that ϕ+ = ϕ + ∈ L!1+ (R, E). In the same way, one proves that each ϕ− ∈ L!1 (R, E), which admits a continuous extension to H− ∪ {∞}, which is holomorphic in H− and vanishes at infinity, belongs to L1− (R, E). The proofs of the “only if” parts of (i) and (ii) are similar. We therefore restrict ourselves to part (i). First we prove the following weaker statement: (i ) Each function ϕ ∈ L!1+ (R, E) admits a continuous extension to H+ ∪ {∞} which is holomorphic in H+ and which satisfies the estimate sup ϕ(λ) ≤ ϕ L!1 (R,E) .

(8.7.15)

λ∈H+

To prove (i ), we first note that, for each u ∈ C00 + (R, E) and λ ∈ H+ ,

! u(λ) ≤

∞

iλx e u(x) dx =

0

∞

−x Im λ e u(x) dx ≤ u

0

L1 (R,A) .

(8.7.16)

Now let an arbitrary function ϕ ∈ L!1+ (R, E) be given, and let f ∈ L1+ (R, E) be the element with ϕ = f!. Choose a sequence uν ∈ C00 + (R, E) with lim f − uμ L1 (R,E) = 0.

μ→∞

(8.7.17)

310


Then lim uμ − uν L1 (R,E) = 0.

μ,ν→∞

(8.7.18)

Recall that ϕ − u !μ L!1 (R,E) = f − uμ L1 (R,E) , by definition of the norm in L!1 (R, E). Therefore it follows from (8.7.17) and (8.7.9) that !μ (λ) = 0. lim sup ϕ(λ) − u

μ→∞ λ∈R

(8.7.19)

From (8.7.18) and (8.7.16) it follows that sup ! uμ (λ) − u !ν (λ) = 0.

lim

μ,ν→∞

λ∈H+

Since the functions u !μ are holomorphic on C, this implies that the sequence u !ν converges uniformly on H+ to some continuous function which is holomorphic in H+ . By (8.7.19), on R, this function coincides with ϕ. So it is proved that ϕ admits a continuous extension to H+ which is holomorphic in H+ . We denote this extension also by ϕ. Moreover, since, again by (8.7.16) sup ! uμ (λ) ≤ uμ L1 (R,A) ,

λ∈H+

we get also (8.7.17). The proof of statement (i ) is complete. Now we consider a function u of the form a if x ∈ [α, β], u(x) = 0 if x ∈ [α, β],

(8.7.20)

where 0 ≤ α < β < ∞ and a ∈ E. Then, it follows immediately from the definition of the Fourier transformation that, for all λ ∈ R with λ = 0,

β

u !(λ) = a

eiλx dx = α

eiλx x=β eiλβ − eiλα =a . iλ x=α iλ

Since α ≥ 0, this formula shows that u ! admits a continuous extension to C, which can be expressed by the same formula for all λ ∈ C with λ = 0. For λ ∈ H+ with λ = 0, this implies that

! u+ (λ) ≤ a

e−β Im λ + e−α Im λ 2 ≤ a . |λ| |λ|

Hence lim

R→∞

sup λ∈H+ ,|λ|≥R

! u+ (λ) = 0

(8.7.21)


311

for each function u of the form (8.7.20). Now let M be the space of all finite sums of functions of the form (8.7.20). Then, for each u ∈ M, u ! is holomorphic on C, and it follows from (8.7.21) that lim

R→∞

! u(λ) = 0

sup

for each u ∈ M.

(8.7.22)

λ∈H+ ,|λ|≥R

Finally let an arbitrary function ϕ ∈ L!1+ (R, E) be given. By statement (i ) we already know that ϕ admits a continuous extension to H+ which is holomorphic in H+ and which satisfies the estimate (8.7.15). It remains to prove that

ϕ(λ) = 0.

sup

lim

R→∞

(8.7.23)

λ∈H+ ,|λ|≥R

Let f ∈ L1+ (R, E) be the function with ϕ = f!. Since M is dense in L1+ (R, E), then we can find a sequence uν ∈ M with lim uν − f L1 (R,A) = 0. As ν→∞

! uν − ϕ L!1 (R,E) = uν − f L1 (R,E) , then it follows from (8.7.15) that lim sup ! uν (λ) − ϕ(λ) = 0,

ν→∞

λ∈H+

and from (8.7.22) we get lim

R→∞

! uν (λ) = 0 for all ν.

sup λ∈H+ ,|λ|≥R

Together this implies (8.7.23).

We conclude this section by computing the Fourier transform of some special functions, which we need in sections 8.9 and 8.10 below. 8.7.8. For all n ∈ N we define xn e−x for x ≥ 0, + θn (x) = 0 for x < 0,

and

θn− (x)

−xn ex = 0

for x ≤ 0, for x > 0.

(8.7.24)

Note that θn+ ∈ L1+ (R, C) ∩ L2 (R, C) and θn− ∈ L1− (R, C) ∩ L2 (R, C). 8.7.9 Lemma. θn+ = (n + 1)! θ0+ ∗ . . . ∗ θ0+ and θn− = (n + 1)! θ0− ∗ . . . ∗ θ0− , + ,. + ,. n+1 times

n+1 times

n ∈ N∗ .

312


Proof. We have for all n ∈ N, ∞ + + + + θ0 (y − x)θn (x)dx = (θn ∗ θ0 )(y) = −∞

−y

y

ex−y xn e−x dx

0

y

=e

xn dx = e−y

0

y n+1 1 = θ+ (y), n+1 n + 1 n+1

and (θn− ∗ θ0− )(y) =

∞

−∞

θ0− (y − x)θn− (x)dx =

0

ey−x xn ex dx y

0

xn dx = −ey

= ey y

y n+1 1 = θ− (y), n+1 n + 1 n+1

+ − i.e., θn+1 = (n + 1)θn+ ∗ θ0+ and θn+1 = (n + 1)θn− ∗ θ0− which implies the assertion by induction.

8.7.10. If ξ ∈ H+ , then Re iξ < 0 and therefore, setting if x ≤ 0, ie−iξx , (x) = ϑ− ξ 0 if x > 0, 1 then we get a function ϑ− ξ ∈ L− (R, C). If ξ ∈ H− , then Re iξ > 0 and therefore, setting if x ≥ 0, −ie−iξx , ϑ+ ξ (x) = 0 if x < 0, 1 then we get a function ϑ+ ξ ∈ L+ (R, C). Note that

θ0+ = iϑ+ −i

and

θ0− = iϑ− i ,

(8.7.25)

where θ0± are the functions introduced in Section 8.7.8. 8.7.11 Lemma.

(i) We have ϑ!− (λ) 1 = !ξ+ λ−ξ ϑ (λ) ξ

if ξ ∈ H+ , if ξ ∈ H− ,

λ ∈ R.

(ii) For all n ∈ N∗ , we have n λ−i (−2)n !+ = (λ) , λ ∈ R, −1 θ λ+i n! n−1 n λ+i 2n !− = (λ) , λ ∈ R. −1 θ λ−i n! n−1

8.8. The Fourier isometry U of L2 R, H

313

Proof. If ξ ∈ H+ , then ϑ!− ξ (λ) =

∞

−∞

eiλx ϑ− ξ (x)dx

0 (iλ−iξ)x

=i

e −∞

x=0 e(iλ−iξ)x 1 dx = i = , iλ − iξ x=−∞ λ−ξ

and if ξ ∈ H− , then ϑ!+ ξ (λ) =

∞

−∞

eiλx ϑ+ ξ (x)dx

= −i

∞

(iλ−iξ)x

e 0

x=∞ e(iλ−iξ)x 1 dx = −i = . iλ − iξ x=0 λ−ξ

This proves part (i). Setting ξ = ±i in part (i), we get i = θ!0+ (λ) λ+i

and

i = θ!0− (λ). λ−i

Hence 2i λ−i −1=− = −2θ!0+ (λ) λ+i λ+i

and

λ+i 2i −1= = 2θ!0− (λ). λ−i λ−i

By Lemma 8.7.9 this yields part (ii).

8.8

The Fourier isometry U of L2 R, H

In this section H is a separable Hilbert space with the scalar product ·, · = ·, ·H . 8.8.1. The space L2 (R, H) introduced in Section 8.7.1, then will be considered a Hilbert space with the scalar product ·, ·L2 (R,H) defined as follows: For functions f, g ∈ C00 (R, H), we set ∞

f (x), g(x) H dx. f, gL2 (R,H) = −∞

If f and g are two arbitrary elements of L2 (Γ, H), then a function f, gH ∈ L1 (R, C) is well defined. Indeed, let {fn }n∈N and {gn }n∈N be two Cauchy sequences of functions from C 0 (R, H) which represent f and g, respectively. Then, by the Cauchy-Schwarz inequality in H, pointwise on R we have

fn , gn H − fm , gm H ≤ fn − fm H gn H + fm H gn − gm H , which implies, by the Cauchy-Schwarz inequality in L2 (R, C), that

fn , gn H − fm , gm H 1 L (Γ,C) ≤ fn − fm L2 (Γ,H) gn L2 (Γ,H) + fm L2 (Γ,H) gn − gm L2 (Γ,H) .

314


Hence fn , gn H n∈N is a Cauchy sequence in L1 (Γ, C), and the function f, gH can be defined to be the limit of this sequence in L1 (Γ, C). We now define the scalar product in L2 (R, H) by

f, gL2 (Γ,H) =

∞

−∞

f, gH dx ,

f, g ∈ L2 (Γ, H).

Finally we note that, for each f ∈ L2 (Γ, H), the function

f H :=

4 f, f H

2 belongs to L (Γ, C) and that, for the norm corresponding to the scalar product f, gL2 (Γ,H) , we have ∞

f 2L2 (R,H) =

f 2H dx. −∞

8.8.2. Recall also the subspaces L2+ (R, H) and L2− (R, H) introduced in Section 8.7.3, and note that (obviously) now the direct sum (8.7.3) is even orthogonal, i.e., we have the orthogonal decomposition L2 (R, H) = L2+ (R, H) ⊕ L2− (R, H). 8.8.3. Let (en )∞ n=1 be an orthonormal basis of H, and let Hn be the subspace of H spanned by en . Then L2 (R, H) splits into the orthogonal sum L2 (R, H) =

∞ =

L2 (R, Hn ).

(8.8.1)

n=1

Therefore each function f ∈ L2 (R, H) has a uniquely determined representation in the form ∞ fn en with fn ∈ L2 (R, C) (8.8.2) f= n=1

and

f 2L2 (R,H) =

∞

fn 2L2 (R,C) .

(8.8.3)

n=1

8.8.4 Proposition. Let f ∈ C00 (R, H). Then the series (8.8.1) converges uniformly. >n Proof. We denote by Pn the orthogonal projector from H onto j=1 Hj . Let ε > 0. Then, for each fixed x ∈ R, there exists nε (x) ∈ N with

f (x) − Pn f (x) ≤

ε 2

for n ≥ nε (x).

8.8. The Fourier isometry U of L2 R, H

315

As f is continuous and Pn = 1, this implies that each x ∈ R has a neighborhood U (x) such that, for y ∈ U (x) and n ≥ nε (x),

f (y) − Pn f (y) ≤ f (y) − f (x) + f (x) − Pn f (x) + Pn f (x) − Pn f (y) ≤ 2 f (y) − f (x) + f (x) − Pn f (x) ≤ ε.

Since the support of f is compact, this completes the proof.

8.8.5. The Fourier transform on L2 (R, H). As, by the preceding proposition, the series (8.8.1) converges uniformly if f ∈ C00 (R, H), for the Fourier transform (Section 8.7.4) f! of a function f ∈ C00 (R, H), it follows from (8.8.1) that f!(λ) =

∞

eiλx dx =

−∞

∞ n=1

∞

−∞

∞ eiλx fn dx en = f!n (λ) en ,

λ ∈ R.

n=1

By (8.8.3) this implies that

f! L2 (R,C) =

∞

f!n L2 (R,H)

for all f ∈ C00 (R, C).

n=1

As, by the Plancherel theorem ([Ru], theorem 9.13), f!n L2 (R,C) = fn L2 (R,C) for all n, this further implies (again using (8.8.3))

f! L2 (R,H) = f L2 (R,H)

for all f ∈ C00 (R, H).

Since, by definition, C00 (R, H) is dense in L2 (R, H), it follows that the Fourier transformation extends to an isometry of L2 (R, H), which we denote by U. For f ∈ L2 (R, H) we write again f! = Uf . 8.8.6. Convolution between L1 R, L(H) and L2 (R, H). First let K ∈ C00 R, L(H) and f ∈ C00 (R, H). Then we define (as usual) (K ∗ f )(y) =

∞

−∞

K(x)f (y − x)dx =

∞

−∞

K(y − x)f (x)dx,

and it follows from Fubini’s theorem that ∞ ∞ (K ∗ f )(λ) = eiλy K(y − x)f (x)dx dy −∞ −∞ ∞ ∞ = eiλ(y−x) K(y − x) dy eiλx f (x)dx −∞ −∞ ∞ ∞ ! f!(λ) , = eiλ(y) K(y) dy eiλx f (x)dx = K(λ) −∞

−∞

y ∈ R,

(8.8.4) λ ∈ C.

316


Since the Fourier transformation is an isometry of L2 (R, H), this implies ! ! f! 2 !

K ∗ f L2 (R,H) = K L (R,H) ≤ sup K(λ)

f L2 (R,H) λ∈R

and further, by (8.7.9) and (8.7.8),

K ∗ f L2 (R,H) ≤ K L1 (R,L(H)) f L2 (R,H) . Since, by definition, C00 (R, H) is dense in L2 (R, H) and C00 R, L(H) is dense in L1 R, L(H) , this implies that the convolution extends to a continuous bilinear map ∗ : L1 R, L(H) × L2 (R, H) −→ L2 (R, H) (called convolution), where, for all K ∈ L1 R, L(H) and f ∈ L2 (R, H),

K ∗ f L2 (R,H) ≤ K L1 (R,L(H)) f L2 (R,H)

(8.8.5)

! f!. K ∗f =K

(8.8.6)

and, by (8.8.4), . This convolution is associative: 8.8.7 Proposition. If K, L ∈ L1 R, L(H) and f ∈ L2 (R, H), then (K ∗ L) ∗ f = K ∗ (L ∗ f ). Proof. If K, L, f are continuous and with compact support, then this follows by Fubini’s theorem: ∞

(K ∗ L) ∗ f (y) = (K ∗ L)(y − x)f (x) dx −∞ ∞ ∞ = K(t)L(y − x − t) dt f (x) dx −∞ −∞ ∞ ∞ = K(t) L(y − x − t)f (x) dx dt −∞ −∞ ∞

K(t)(L ∗ f )(y − t) dt = K ∗ (L ∗ f ) (y) , y ∈ R. = −∞

By continuity (see estimate (8.8.5)), this implies that the assertion holds also in the general case. 2 8.8.8 Proposition. Let P+ be the orthogonal projector from L (R, H) onto 2 2 1 L+ (R, H). Then, for all f ∈ L (R, H), K+ ∈ L+ R, L(H) and K− ∈ L1− R, L(H) ,

(8.8.7) K+ ∗ (P+ f ) = P+ K+ ∗ (P+ f ) ,

P+ K− ∗ f = P+ K− ∗ (P+ f ) . (8.8.8)

8.9. The isometry V from L2 (T, H) onto L2 (R, H)

317

Proof. In view of (8.8.5), we may assume that the functions f, K+ and K− are piecewise continuous. Then ∞ y

K+ ∗ (P+ f ) (y) = K+ (y − x)(P+ f )(x) dx = K+ (y − x)f (x) dx −∞

0

L2+ (R, H),

for all y ∈ R. This implies that K+ ∗ (P+ f ) ∈ Moreover ∞

P+ K− ∗ f (y) = K− (y − x)f (x) dx = −∞

y

∞

i.e., we have (8.8.7).

K− (y − x)f (x) dx

for all y ∈ R. For y ≥ 0, this implies that ∞

P+ K− ∗ f (y) = K− (y − x) P+ f (x) dx = P+ K− ∗ (P+ f ) (y), y

i.e., we have (8.8.8).

8.9

The isometry V from L2 (T, H) onto L2 (R, H)

In this section, H is again a separable Hilbert space, and L2 (R, H), L2+ (R, H), L2− (R, H) are the Hilbert spaces introduced in Section 8.8.1. Recall the (obvious) orthogonal decomposition L2 (R, H) = L2+ (R, H) ⊕ L2− (R, H).

(8.9.1)

Further, in this section, T is the unit circle, and L (T, H), are the Hilbert spaces introduced in Section 8.3.1 and in Theorem and Definition 8.3.10. Recall that, by part (iii) of Theorem and Definition 8.3.10, we have the orthogonal decomposition 2

L2+ (T, H),

L2− (T, H)

L2 (T, H) = L2+ (T, H) ⊕ L2− (T, H).

(8.9.2)

8.9.1. Let Φ be the Möbius transform defined by Φ(z) = i

1+z 1−z

if λ ∈ C \ {1},

Φ(∞) = −i and

Φ(1) = ∞.

Note that then λ−i if λ ∈ C \ {−i}, Φ−1 (−i) = ∞ and Φ−1 (∞) = 1. Φ−1 (λ) = λ+i Further, let C 0 T \ {1}, H and C 0 (R, H) be the spaces of continuous H-valued functions defined on T \ {1} and R, respectively. Set (Vf )(λ) =

Φ−1 (λ) − 1 −1 √ f Φ (λ) 2

for f ∈ C 0 T \ {∞} and λ ∈ R. (8.9.3)

As Φ is a diffeomorphism from T \ {1} onto R, in this way a linear isomorphism V from C 0 T \ {1}, H onto C 0 (R, H) is defined.

318


It is the aim of this section to study this linear isomorphism V. We begin with the following 8.9.2 Theorem and Definition. For all f ∈ C 0 T \ {1}, H , ∞

f 2 |dz| =

(Vf )(λ) 2 dλ. (8.9.4) T

−∞

Since (by our definitions of L (T, H) and L2 (R, H)), the space L2 (T, H)∩C 0 (T, H) is dense in L2 (T, H), and the space L2 (R, H) ∩ C 0 (R, H) is dense in L2 (R, H), it 2 0 follows from (8.9.4) that the restriction of V to L (T, H) ∩ C T \ {1}, H extends to a linear isometry from L2 (T, H) onto L2 (R, H). We denote this isometry also by V. Moreover, also the isometry from L2 (T, C) onto L2 (R, C) (obtained for H = C) will be denoted by V. 2

Proof. Set

1 + eit ϕ(t) = Φ eit = i 1 − eit

for 0 < t < 2π.

As Φ is a C ∞ -diffeomorphism from T \ {1} onto R and the map t → eit is a C ∞ diffeomorphism from ]0, 2π[ onto T \ {1}, this is a C ∞ -diffeomorphism from ]0, 2π[ onto R. We have ϕ (t) = i

ieit (1 − eit ) + (1 + eit )ieit eit = −2 , (1 − eit )2 (1 − eit )2

t ∈]0, 2π[.

(8.9.5)

Set ψ = ϕ−1 . Then

λ = ϕ ψ(λ) = Φ eiψ(λ) , and therefore

Φ−1 (λ) = eiψ(λ) ,

λ ∈ R, λ ∈ R.

(8.9.6)

Further, by (8.9.5), 2 1 − eiψ(λ) 1 ψ (λ) = − , = −2 ϕ (ψ(λ)) eiψ(λ)

λ ∈ R.

Together with (8.9.6) this implies that 2 1 − Φ−1 (λ) ψ (λ) = − , λ ∈ R. (8.9.7) 2Φ−1 (λ) Now let a function f ∈ C 0 T \ {1}, H be given. Then, by definition of |dz|,

f (z) |dz| = 2

T

0

2π

f eit 2 dt.

8.9. The isometry V from L2 (T, H) onto L2 (R, H)

319

Since ψ is a diffeomorphism from R onto ]0, 2π[ and by (8.9.6) and (8.9.7), this yields ∞ iψ(λ) 2 ψ (λ) dλ f e

f (z) 2 |dz| = T −∞ ∞ −1 2 1 − Φ−1 (λ) 2 f Φ (λ) = dλ. 2Φ−1 (λ) −∞ As Φ−1 (λ) = 1 for all λ ∈ R, this further implies that T

f (z) 2 |dz| =

∞

−∞

1 − Φ−1 (λ) −1 2 dλ. √ (λ) f Φ 2

By definition of V this is (8.9.4).

8.9.3. In what follows the functions ωn+ ∈ L2+ (T, C), n ∈ N, and ωn− ∈ L2− (T, C), n ∈ N, defined by n 1 1 + n − ωn (z) = (z − 1) and ωn (z) = − −1 , z z play a special role. Recall that, by part (iii) of Theorem and Definition 8.3.10, the linear space spanned by the functions z n , n ∈ N, is a dense subspace of L2+ (T, H), and the linear space spanned by the functions 1/z n , n ∈ N∗ , is a dense subspace of L2− (T, H). As each of the functions z n , n ∈ N, is a linear combination of some of the functions ωn+ , n ∈ N, and each of the functions 1/z n , n ∈ N∗ , is a linear combination of some of the functions ωn− , n ∈ N, this implies: The linear space spanned by the functions ωn+ , n ∈ N, is a dense subspace of 2 L+ (T, C), and the linear space spanned by the functions ωn− , n ∈ N∗ , is a dense subspace of L2− (T, C). In view of the orthogonal decomposition (8.9.2), this further implies: Then linear space spanned by the functions ωn+ and ωn− , n ∈ N, is a dense subspace of L2 (T, H). Further note that, by definition of V, for all n ∈ N, n+1 λ−i 1 Vωn+ (λ) = √ , λ ∈ R, −1 2 λ+i (8.9.8) n+1 λ+i 1 − Vωn (λ) = √ , λ ∈ R, −1 2 λ−i and therefore, by lemma (8.7.11) (ii), (−2)n+1 !+ Vωn+ = √ θn 2(n + 1)!

and Vωn− = √

2n+1 θ!n− . 2(n + 1)!

(8.9.9)

320


8.9.4 Theorem. Let θn+ ∈ L2+ (R, C) and θn− ∈ L2− (R, C), n ∈ N, be the functions introduced in Section 8.7.8. Then the linear space spanned by the functions θn+ , n ∈ N, is dense in L2+ (R, C), and the linear space spanned by the functions θn− , n ∈ N, is dense in L2− (R, C). Proof. Since the linear space spanned by the functions ωn+ and ωn− , n ∈ N, is dense in L2 (T, C) (Section 8.9.3), it follows from Theorem and Definition 8.9.2 that the linear space spanned by the functions Vωn+ and Vωn− , n ∈ N, is dense in L2 (R, C). By (8.9.9) this implies that the linear space spanned by the functions θ!n+ and θ!n− , n ∈ N, is dense in L2 (R, C). As the Fourier transformation is an isometry of L2 (R, C), this further implies that the linear space spanned by the functions θn+ and θn− , n ∈ N, is dense in L2 (R, C). Taking into account the orthogonal decomposition (8.9.1) and the fact that θn+ ∈ L2+ (R, C) and θn− ∈ L2− (R, C), this completes the proof. 8.9.5 Theorem. Let P T be the orthogonal projector from L2 (T, H) onto L2+ (T, H), let P R be the orthogonal projector from L2 (R, H) onto L2+ (R, H) and let U be the Fourier isometry of L2 (R, H) (Section 8.8.5). Then P R = U−1 VP T V−1 U.

(8.9.10)

Proof. It follows from (8.9.9) that, for all n ∈ N, (−2)n+1 + U−1 Vωn+ = √ θn 2(n + 1)!

and

U−1 Vωn− = √

2n+1 θn− . 2(n + 1)!

(8.9.11)

As the decomposition (8.9.2) is orthogonal, we have P T ωn+ = ωn+ and P T ωn− = 0 for all n ∈ N. Therefore it follows from (8.9.11) that, for all n ∈ N, U−1 VP T V−1 Uθn+ = θn+

and

U−1 VP T V−1 Uθn− = 0.

Taking into account Theorem 8.9.4 and the fact that also the decomposition (8.9.1) is orthogonal, this implies (8.9.10).

8.10

The algebra functions of operator 1 L(H) ⊕ L R, L(H)

In this section, H is again a Hilbert space. 8.10.1. Let L!1 R, L(H) , L!1+ R, L(H) and L!1− R, L(H) be the Banach algebras introduced in Section 8.7.5. Then we denote by L(H) ⊕ L!1 R, L(H) the algebra of functions W : R → L(H) of the form ! W (λ) = A + K(λ),

λ ∈ R,

(8.10.1)

8.10. The algebra of operator functions L(H) ⊕ L1 R, L(H)

321

! =0 where K ∈ L1 R, L(H) and A ∈ L(H) is a constant operator. Since K(∞) 1 for all K ∈ L R, L(H) (Theorem 8.7.7 (iii)), the representation of a function W ∈ L(H) ⊕ L!1 R, L(H) in the form (8.10.1) is uniquely determined. Therefore we can introduce a norm · L(H)⊕L!1 (R,L(H)) in L(H) ⊕ L!1 R, L(H) , setting ! ! !1

A + K !1 (R,L(H)) = A L(H) + K L (R,L(H)) L(H)⊕L for A ∈ L(H) and K ∈ L1 R, L(H) . Note that then, by definition of the norm in L!1 R, L(H) (see (8.7.8)), !

A + K !1 (R,L(H)) = A L(H) + K L1 (R,L(H)) L(H)⊕L for all A ∈ L(H) and K ∈ L1 R, L(H) . It is easy to see that in this way, L(H) ⊕ L!1 R, L(H) becomes a Banach space and that, for all V, W ∈ L(H) ⊕ L!1 R, L(H) ,

V W L(H)⊕L!1 (R,L(H)) ≤ V L(H)⊕L!1 (R,L(H)) W L(H)⊕L!1 (R,L(H)) .

(8.10.2)

Hence, with this norm, L(H) ⊕ L!1 (R, L(H)) is a Banach algebra. As L!1+ (R, L(H)) and L!1− (R, L(H)) are subalgebras of L!1 (R, L(H)), it follows that L(H) ⊕ L!1+ (R, L(H)) and L(H) ⊕ L!1( R, L(H)) are subalgebras of this Banach algebra. We identify L(H) with the subalgebra of L(H) ⊕ L!1 (R, L(H)) which consists of the constant L(H)-valued functions. As L!1 R, L(H) splits into the direct sum L!1 R, L(H) = L!1− (R, L(H)) ⊕ L!1+ (R, L(H)), then L(H) ⊕ L!1 (R, L(H)) splits into the direct sum L(H) ⊕ L!1 (R, L(H)) = L(H) ⊕ L!1− (R, L(H)) ⊕ L!1+ (R, L(H)).

(8.10.3)

From Theorem 8.7.7 we immediately get the following 8.10.2 Theorem. (i) A function W ∈ L(H) ⊕ L!1 R, L(H) belongs to L(H) ⊕ L!1+ R, L(H) , if and only if it admits a continuous extension to H+ ∪ {∞} which is holomorphic in H+ , and then max

λ∈H+ ∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

(ii) A function W ∈ L(H)⊕ L!1 R, L(H) belongs to L(H)⊕ L!1− R, L(H) , if and only if it admits a continuous extension to H− ∪ {∞} which is holomorphic in H− , and then max

λ∈H− ∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

322


(iii) Each function W ∈ L(H)⊕L!1 R, L(H) is continuous on R∪{∞}. Moreover, for each W ∈ L(H) ⊕ L!1 R, L(H) , max λ∈R∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

The main result of the present section is the following 8.10.3 Theorem. The space of L(H)-valued rational functions without poles on R ∪ {∞} is contained in L(H) ⊕ L!1 (R, L(H)) as a dense subset. To prove this, we first deduce from Theorem 8.9.4 the following 8.10.4 Lemma. Let θn± , n ∈ N, be the functions introduced in Section 8.7.8. Then the linear space spanned by the functions θn+ , n ∈ N, is dense in L1+ (R, C), and the linear space spanned by the functions θn− , n ∈ N, is dense in L1− (R, C). Proof. It is sufficient to prove that the functions θn+ , n ∈ N, span a dense subspace of L1+ (R, C), because the statement with respect to the functions θn− then follows by the substitution x → −x. Let f : [0, ∞[→ C be an arbitrary bounded measurable function with ∞ xn e−x f (x) dx = 0 for all n ∈ N. (8.10.4) 0

By the Hahn-Banach theorem it is sufficient to prove that then f ≡ 0. Set h(x) = e−x f (2x),

x ≥ 0.

Then h belongs to L2+ (R, C), and, with the substituion x → x/2, it follows from (8.10.4) that, for all n ∈ N, ∞ ∞ ∞ 1 xn e−x h(x)dx = xn e−2x f (2x)dx = n+1 xn e−x f (x)dx = 0. 2 0 0 0 Since, by Theorem 8.9.4, the space spanned by the functions θn+ is dense in L2+ (R, C), this implies that h ≡ 0. Hence f ≡ 0. 8.10.5. Proof of Theorem 8.10.3. It is sufficient to prove this for H = L(H) = C. By Lemma 8.7.11 (i), for each ξ ∈ C \ R, the function 1 λ−ξ belongs to L!1 (R, C). Since C ⊕ L!1 (R, C) is an algebra and since also the constant functions belong to this algebra, this implies that C⊕ L!1 (R, C) contains all rational functions without poles on R ∪ {∞}. It remains to prove the density. By Lemma 8.10.4, the space spanned by the functions θn− and θn+ , n ∈ N, is dense in L1 (R, C). Hence (by definition of the norm

8.10. The algebra of operator functions L(H) ⊕ L1 R, L(H)

323

in L!1 (R, C)) the space spanned by the functions θ!n− and θ!n+ , n ∈ N, is dense in L!1 (R, C). By part (ii) of Lemma 8.7.11, this implies that the space spanned by the functions n n λ−i λ+i and n ∈ N∗ , (8.10.5) −1 −1 , λ+i λ−i is dense in L!1 (R, C). Hence the space spanned by the functions (8.10.5) and the constant functions is dense in C ⊕ L!1 (R, C). Since all these functions are rational and without poles on R ∪ {∞}, this completes the proof. Using again the M¨ obius transformation (λ − i)/(λ + i) (Section 8.9.1), we now obtain the following version of Theorem 8.10.1: 8.10.6 Theorem. Let W ∈ L(H) ⊕ L!1 R, L(H) such that W (λ) ∈ GL(H) for all λ ∈ R ∪ {∞}. Then: (i) The pointwise defined function W −1 again belongs to L(H) ⊕ L!1 R, L(H) . (ii) The function W can be written in the form W = V − V V+ , where V− : H− ∪{∞} → GL(H) is continuous on H− ∪{∞} and holomorphic in H− ∪ {∞}, V+ : H+ → GL(H) is continuous on H+ and holomorphic in H+ , V is an L(H)-valued rational function without poles on R ∪ {∞}, V (λ) ∈ GL(H) for all λ ∈ R ∪ {∞}, and the functions V− , V−−1 , V+ and V+−1 belong to L(H) ⊕ L!1 R, L(H) . 7 ∈ L(H) ⊕ L!1 R, L(H) be a second function with W (λ) ∈ GL(H) for (iii) Let W all λ ∈ R ∪ {∞}, and assume that 7 W+ W = W− W

on R ∪ {∞},

where W− : H− ∪ {∞} → GL(H) and W+ : H+ → GA are continuous functions which are holomorphic in H− ∪ {∞} and H+ , respectively. Then W− , W−−1 ∈ L(H) ⊕ L!1− R, L(H) and W+ , W+−1 ∈ L(H) ⊕ L!1− R, L(H) . Proof. Let Φ be the M¨ obius transform introduced in Section 8.9.1, let T be the unit disc, let D+ be the unit circle and let D− := C \ D+ . We denote by R the Banach algebra of all operator functions A : T → L(H) of the form A = W ◦ Φ with W ∈ L(H) ⊕ L!1 R, L(H) , endowed with the norm

A R := W L(H)⊕L!1 (R,L(H)) . Since Φ maps T onto T ∪ {∞}, D+ onto H+ and D− onto H− , then it follows from Theorem 8.10.2 (iii), Theorem 8.10.3 and the decomposition (8.10.3) that R satisfies conditions (A), (B), (C) from Section 7.2.4. Therefore, we can apply Theorem 7.2.5 to R, and we obtain:

324


(i ) W −1 ◦ Φ ∈ L(H) ⊕ L!1 R, L(H) . (ii ) The function W ◦ Φ can be written in the form W ◦ Φ = V−T V T V+T , where V−T : D− ∪{∞} → GL(H) is continuous on D− ∪{∞} and holomorphic in D− ∪ {∞}, V+T : D+ → GL(H) is continuous on D+ and holomorphic in D+ , V T is an L(H)-valued rational function without poles on T, V T (z) ∈ GL(H) for all z ∈ T, and the functions V−T , (V−T )−1 , V+T and (V+T )−1 belong to R. (iii ) The functions W− ◦ Φ, W−−1 ◦ Φ, W+ ◦ Φ and W+−1 ◦ Φ belong to R. Then (i) and (iii) follow from (i ) and (iii ) by definition of R, and (ii) follows from (ii ) setting V± := V±T ◦ Φ−1 and V := V T ◦ Φ−1 .

8.11

Factorization with respect to the real line

Throughout this section, E is a Banach space. Here we introduce the notion of factorization with respect to the real line, and we show that this is equivalent to the notion of factorization with respect to the unit circle. 8.11.1 Definition. Let G be one of the groups GL(E), G ∞ (E) or G ω (E) (Def. 5.12.1), and let A : R ∪ {∞} → G be a continuous function (cf. Section 8.7.6). A representation of A in the form A = A− ΔA+

(8.11.1)

will be called a factorization of A relative to R and G if the factors A− , A+ , Δ have the following properties: – Either Δ ≡ I or Δ is of the form Δ(λ) = P0 +

κ n λ−i j j=1

λ+i

Pj ,

λ ∈ R,

(8.11.2)

where n ∈ N∗ , κ1 > . . . > κn are non-zero integers, P1 , . . . , Pn are mutually disjoint finite dimensional projectors in E, and P0 = I − P1 − . . . − Pn ; – A+ is a continuous GL(E)-valued function on H+ ∪ {∞}, which is holomorphic in H+ ; – A− is a continuous GL(E)-valued function on H− ∪ {∞}, which is holomorphic in H− . If Δ ≡ I, then this factorization will be called canonical .

8.12. Wiener-Hopf integral operators in L2 [0, ∞[, H

325

8.11.2. Let Φ : C ∪ {∞} → C ∪ {∞} be the M¨ obius transform defined by Φ(z) = i

1+z 1−z

(cf. Section (8.9.1), let T be the unit circle, let D+ be the open unit disc, and let D− := C \ D+ . Since ΦD is a homeomorphism from D+ onto H+ ∪ {∞}, + which is biholomorphic from D+ onto H+ , and Φ is a homeomorphism D − ∪{∞}

from D− ∪ {∞} onto H− ∪ {∞}, which is biholomorphic from D− ∪ {∞} onto H− , we get the following simple but important 8.11.3 Proposition. Let G be one of the groups GL(E), G ∞ (E) or G ω (E) (Def. 5.12.1), and let A : R ∪ {∞} → G be a continuous function (cf. Section 8.7.6). Then A = A− ΔA+ is a factorization of A relative to R and G, if and only if, A ◦ Φ = A− ◦ Φ Δ ◦ Φ A+ ◦ Φ is a factorization of A ◦ Φ relative to T and G. 8.11.4 Definition. In view of the corresponding fact for factorizations with respect to T (see Section 7.1.2), this proposition implies that the integers κ1 , . . . , κn and the dimensions of the projectors P1 , . . . , Pn in Definition 8.11.1 are uniquely determined by A. The integers κ1 , . . . , κn will be called the non-zero partial indices of A, and the number dim Pj will be called the multiplicity of κj . Moreover, Proposition 8.11.3 shows that, for each factorization result relative to T, there is a corresponding factorization result with respect to the real line. In the following sections we use this fact to study the Wiener-Hopf integral equation on the half line.

8.12

Wiener-Hopf integral operators in L2 [0, ∞[, H

Let H be a separable Hilbert space. In this section we use without further reference the notations introduced in sections 8.7.1, 8.8.1, and 8.8.6. Here we study the Wiener-Hopf integral equation ∞ K(y − x)f (x) dx = f (y), y ≥ 0, (8.12.1) u(y) − −∞

1

where K ∈ L R, L(H) and f ∈ L2+ (R, H) are given, and u ∈ L2 (R, H) is sought. Throughout this section we moreover use without further reference the notations and facts given in the following Section 8.12.1.

326


8.12.1. Let P R be the orthogonal projector from L2 (R, H) onto L2+ (R, H). Then Ker P R = L2− (R, H), because of the (obvious) orthogonal decomposition L2 (R, H) = L2+ (R, H) ⊕ L2− (R, H).

(8.12.2)

If K ∈ L1 R, L(H) , then the operator T acting in L2+ (R, H) by T u = u − P R (K ∗ u),

u ∈ L2+ (R, H),

is called the Wiener-Hopf operator with kernel function K. With a Wiener-Hopf operator with kernel function K we moreover associate the symbol of this WienerHopf operator, which is defined to be the operator function ! I − K.

(8.12.3)

! of the kernel function K and the Then we speak also about the symbol I − K ! kernel function K of the symbol I − K The set of all symbols with kernel function in L1 R,L(H) , i.e., the set of all functions of the form (8.12.3) with K ∈ L1 R, L(H) will be denoted by S R, L(H) . If L(H) ⊕ L!1 R, L(H) is the Banach algebra from Section 8.10, then S R, L(H) = W ∈ L(H) ⊕ L!1 R, L(H) W (∞) = I .

(8.12.4)

The Wiener-Hopf operator with symbol W ∈ S R, L(H) will be denoted by TW . So, if K ∈ L1 (R, H) and W is the symbol of K, then equation (8.12.1) takes the form TW u = f . of S R, L(H) We denote by S− R, L(H) and S+ R, L(H) the subsets which consist of the symbols with kernel function in L−1 R,L(H) and L+1 R,L(H) , respectively, i.e., S+ R, L(H) = W ∈ L(H) ⊕ L!1+ R, L(H) W (∞) = I , (8.12.5) S− R, L(H) = W ∈ L(H) ⊕ L!1− R, L(H) W (∞) = I . From (8.12.5), we get the following simple 8.12.2 Proposition. Let W ∈ S R, L(H) admit a factorization W = W− ΔW+ relative to R and GL(H) (Def. 8.11.1). Then this factorization can be chosen with W− , W−−1 ∈ S− R, L(H)

and

W+ , W+−1 ∈ S+ R, L(H) .

(8.12.6)

If W admits a canonical factorization, then the canonical factorization with (8.12.6) is uniquely determined.


327

Proof. Let ⎛ 7− (λ) ⎝P0 + W (λ) = W

κ n λ−i j j=1

λ+i

⎞ 7+ (λ), Pj ⎠ W

λ ∈ R,

(8.12.7)

be an arbitrary factorization of W . As W (∞) = I, then 7− (∞)W 7+ (∞). I = W (∞) = W 7− (∞), W− = W 7− A−1 , W+ = AW 7+ , and Pj = APj A−1 for 0 ≤ j ≤ n. Set A = W Then W− (∞) = W+ (∞) = I. Therefore, by (8.12.5), we have (8.12.6). Moreover, then we get a factorization ⎞ ⎛ κj n λ − i 7 (λ) ⎝P0 + 7+ (λ) W (λ) = W Pj ⎠ W − λ + i j=1 ⎛ ⎞ κj n λ − i 7− (λ)A−1 ⎝AP0 A−1 + 7+ (λ) APj A−1 ⎠ AW =W λ + i j=1 ⎞ ⎛ κj n λ − i Pj ⎠ W+ (λ), λ ∈ R, = W− (λ) ⎝P0 + λ+i j=1 of the required form. The assertion of uniqueness in the case of canonical factorizations follows from the fact that holomorphic functions on C ∪ {∞} are constant (using first Theorem 1.5.4). Moreover, in view of (8.12.4) and (8.12.5), we immediately obtain the following corollaries of theorems 8.10.2 and 8.10.6: 8.12.3 Corollary. (i) A symbol W ∈ S R, L(H) belongs to S+ R, L(H) , if and only if it admits a continuous extension to H+ ∪ {∞} which is holomorphic in H+ , and then max

λ∈H+ ∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

(ii) A symbol W ∈ S R, L(H) belongs to S− R, L(H) , if and only if it admits a continuous extension to H− ∪ {∞} which is holomorphic in H− , and then max

λ∈H− ∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

328


(iii) Each symbol W ∈ S R, L(H) is continuous on R ∪ {∞}. Moreover, for each W ∈ S R, L(H) , max λ∈R∪{∞}

W (λ) ≤ W L(H)⊕L!1 (R,L(H)) .

8.12.4 Corollary. Let W ∈ S R, L(H) such that W (λ) ∈ GL(H) for all λ ∈ R.9 Then: (i) The pointwise defined function W −1 again belongs to S R, L(H) . (ii) The function W can be written in the form W = V − V V+ , where V− (λ) ∈ GL(H),

λ ∈ H− ∪ {∞},

V+ (λ) ∈ GL(H),

λ ∈ H+ ∪ {∞},

V− , V−−1 ∈ S− R, L(H) , V+ , V+−1 ∈ S+ R, L(H) .

7 ∈ S R, L(H) be a second symbol with W 7 (λ) ∈ GL(H) for all λ ∈ R, (iii) Let W and assume that 7W W = W− W on R, + where W− : H− ∪ {∞} → GL(H) and W+ : H+ ∪ {∞} → GA are continuous functions which are holomorphic in H− and H+ , respectively, and

Then W− , W−−1 ∈ S−

W− (∞) = W+ (∞) = I. R, L(H) and W+ , W+−1 ∈ S+ R, L(H) .

In this section we prove the following three theorems: 8.12.5 Theorem. Let W ∈ S R, L(H) . Then the following two conditions are equivalent: (i) W (λ) = 0 for all λ ∈ R, and W admits a canonical factorization relative to R and GL(H) (Def. 8.11.1). (ii) The Wiener-Hopf operator TW with symbol W is invertible. In that case we have: If W = W− W+ is the canonical factorization of W with W− , W−−1 ∈ S− R, L(H) and W+ , W+−1 ∈ S+ R, L(H) , (8.12.8) (see Proposition 8.12.2), then the inverse of TW is obtained in the following way: If K± ∈ L1± R, L(H) are the kernel functions of W±−1 , i.e., 0 ∞ W−−1 (λ) = I − eiλx K− (x) dx and W+−1 (λ) = I − eiλx K+ (x) dx, −∞

9 Note

0

that then W (λ) ∈ GL(H) for all λ ∈ R ∪ {∞}, as always W (∞) = I.


329

then, for all u ∈ L2+ (R, H), −1 u)(y) (TW

where

K(y, x) =

∞

= u(y) +

K(y, x)u(x) dx,

y ≥ 0,

(8.12.9)

0

⎧ x ⎪ ⎪ ⎨−K+ (y − x) + K+ (y − t)K− (t − x) dt

if 0 ≤ x ≤ y < ∞,

⎪ ⎪ ⎩−K− (y − x) +

if 0 ≤ y ≤ x < ∞.

0 y

K+ (y − t)K− (t − x) dt

0

8.12.6 Theorem. Let W ∈ S R, L(H) . Then the following two conditions are equivalent: (i) W (λ) = 0 for all λ ∈ R, and W admits a factorization relative to R and GL(H) (Def. 8.11.1). (ii) The Wiener-Hopf operator TW with symbol W is a Fredholm operator. In that case we have: If κ n λ−i j W (λ) = W− (λ) P0 + Pj W+ (λ), λ+i j=1

λ ∈ R,

is a factorization of W with respect to R and GL(H), and if r is the index with κ1 > . . . > κr > 0 > κr+1 > . . . > κn , then dim Ker TW = −

r

κj dim Pj

and

dim Coker TW =

j=1

n

κj dim Pj .

j=r+1

(8.12.10) 8.12.7 Theorem. Let W ∈ S R, L(H) be a symbol satisfying the equivalent conditions (i) and (ii) in Theorem 8.12.6, let κ n λ−i j W (λ) = W− (λ) P0 + Pj W+ (λ), λ ∈ R, λ+i j=1

be a factorization of W relative to R and GL(H) with W− , W−−1 ∈ S− R, L(H) and W+ , W+−1 ∈ S+ R, L(H)

(8.12.11)

(which then exists by Proposition 8.12.2), let r be the index with κ1 > . . . > κr > 0 > κr+1 > . . . > κn , and let Δ(λ) := P0 +

κ n λ−i j j=1

λ+i

Pj ,

λ ∈ R.

330


Then: Δ, Δ−1 ∈ S R, L(H) (this holds by (8.12.4)) and (assertion of the theorem) TW −1 TΔ−1 TW −1 +

−

is a generalized inverse (Section 6.10.2) of TW , and the operator TΔ−1 is given by TΔ−1 = I +

n

Tj Pj ,

(8.12.12)

j=1

where, for all u ∈ L2+ (R, H) and y ≥ 0, (Tj u)(y) = (Pj u)(y) −

κj ν κj 2 ν=1

ν

ν!

∞

(y − x)ν−1 ey−x (Pj u)(x)dx

y

(8.12.13)

if r + 1 ≤ j ≤ n, and −κj (−2)ν

−κj

(Tj u)(y) = (Pj u)(y) +

ν=1

ν

ν!

y

(y − x)ν−1 ex−y (Pj )u(x)dx

0

if r + 1 ≤ j ≤ n. (8.12.14) The remainder of this section is devoted to the proof of these three theorems. 8.12.8. Let Φ : C ∪ {∞} → C ∪ {∞} be the M¨ obius transform defined by Φ(z) = i

1+z , 1−z

z ∈ C,

(cf. Section (8.9.1)), let T be the unit circle, and let L2 (T, H), L2+ (T, H), L2− (T, H) be the Hilbert spaces introduced in Section 8.3.1 and in Theorem and Definition 8.3.10. Recall that, by part (iii) of Theorem and Definition 8.3.10, then we have the orthogonal decomposition L2 (T, H) = L2+ (T, H) ⊕ L2− (T, H).

(8.12.15)

Let P T be the orthogonal projector from L2 (T, H) onto L2+ (T, H). Note that then fromthe orthogonal decomposition (8.12.15). Ker P T = L2− (T, H), as it follows For each symbol S ∈ S R, L(H) , together with the Wiener-Hopf operator TS acting in L2 (R, H), we also consider the Wiener-Hopf operator WS◦Φ acting in L2+ (T, H) by

f ∈ L2+ (T, H), WS◦Φ f = P T (S ◦ φ)f , which was studied already in Section 8.4.


331

8.12.9 Proposition. Let V be the isometry introduced in Theorem and Definition 8.9.2, and let U be the Fourier isometry of L2 (R, H) (Section 8.8.5). Then for all S ∈ S R, L(H) . (8.12.16) WS◦Φ = V−1 UTS U−1 V Proof. Let S ∈ S R, L(H) be given. First note that

(S ◦ φ)f = V−1 S(Vf )

for all f ∈ L2 (T, H).

(8.12.17)

Indeed, if f belongs to the linear space C 0 (T, H) ∩ L2 (T, H), this follows from the definition (8.9.3) of V. Since this space is dense in L2 (T, H) (by our definition of L2 (T, H), it follows for all f ∈ L2 (T, H). Now let K be the kernel function of S. Then, for all u ∈ L2+ (R, H), it follows from (8.8.6) and (8.12.17) that ! TS u = P R (u − K ∗ u) = P R U−1 Uu − U(K ∗ u) = P R U−1 Uu − KUu

! = P R U−1 S(Uu) = P R U−1 (I − K)Uu for all u ∈ L2+ (R, H). Now let f ∈ L2 (T, H) be given. Then this implies that

V−1 UTS U−1 Vf = V−1 UP R U−1 S(UU−1 Vf ) = V−1 UP R U−1 S(Vf ) = V−1 UP R U−1 V V−1 S(Vf ) . As, by Theorem 8.9.5, V−1 UP R U−1 V = P T and, by (8.12.17), V−1 S(Vf ) = (S ◦ φ)f , this further implies

V−1 UTS U−1 Vf = P T (S ◦ φ)f , i.e.,

V−1 UTS U−1 Vf = WS◦Φ f.

8.12.10 Lemma. Let W ∈ S R, L(H) such that TW is a Fredholm operator. Then W (λ) is invertible for all λ ∈ R.

Proof. Let Φ be the Möbius transformation introduced in Section 8.9.1. Then, by Proposition 8.12.9, also the Wiener-Hopf operator WW ◦Φ is a Fredholm operator. Therefore the assertion follows from Proposition 8.6.6. 8.12.11. Proof of Theorem 8.12.5. By Lemma 8.12.10 we may already assume that W (λ) ∈ GL(H)

for all λ ∈ R.

(8.12.18)

332


Let Φ(z) := z

1+z . 1−z

As W (∞) = I, then it follows from (8.12.18) that (W ◦ Φ)(z) is invertible for all z ∈ T, where T is the unit circle. Moreover, part (ii) of Corollary 8.12.4 in particular implies that W ◦ Φ admits local factorizations with respect to T and GL(H) (Def. 7.1.3). Therefore, by Theorem 8.4.2, the Wiener-Hopf operator WW ◦Φ is invertible, if and only if, W ◦ Φ admits a canonical factorization relative to T and GL(H). As, by Proposition 8.12.9, WW ◦Φ = V−1 UTW U−1 V, this implies that TW is invertible, if and only if, W ◦ Φ admits a canonical factorization relative to T and GL(H). Finally, by Proposition 8.11.3, this yields that TW is invertible, if and only if, W admits a canonical factorization relative to R and GL(H). So the equivalence of the conditions (i) and (ii) in Theorem 8.12.5 is proved. Now let W = W− W+ be the canonical factorization of W with (8.12.8). As the constant function with value I belongs to S R, L(H) , then part (iii) of Corollary 8.12.4 implies (8.12.8). Hence, by parts (ii) and (iii) of Proposition 8.12.14, the operators TW− and TW+ are invertible, where −1 = TW −1 TW −

−

and

−1 TW = TW −1 , +

+

and, by part (i) of this proposition, T W = T W − T W+ . Together this implies that the inverse of TW is given by −1 = TW −1 TW −1 . TW +

(8.12.19)

−

Let K+ and K− be the kernels of W+−1 and W−−1 , respectively, and let u ∈ L2+ (R, H). Then, by (8.12.19),

−1 u (y) = TW −1 TW −1 u (y) TW + − ∞ K+ (y − x) TW −1 u (x) dx = TW −1 u (y) − − − −∞ ∞

(K− (y − x) + K+ (y − x) u(x)dx = u(y) − −∞ ∞ ∞ + K+ (y − x)K− (x − t) u(t) dt dx, −∞

−∞

(8.12.20)

y ≥ 0.


333

First applying Fubini’s theorem and then changing the role of x and t, we get ∞ ∞ K+ (y − x)K− (x − t) u(t) dt dx −∞ −∞ ∞ ∞ = K+ (y − x)K− (x − t) u(t) dx dt −∞ −∞ ∞ ∞ = K+ (y − t)K− (t − x) dt u(x) dx, y ≥ 0. −∞

−∞

Therefore it follows from (8.12.20) that ∞ −1 TW u (y) = u(y) + L(y, x)u(x)dx, −∞

y ≥ 0,

(8.12.21)

where ∞ L(y, x) := −K− (x − y) − K+ (x − y) +

K+ (y − t)K− (t − x) dt,

y, x ≥ 0.

−∞

Since K+ (s) = 0 if s ≤ 0 and K− (s) = 0 if s ≥ 0, we see that L(y, x) = K(y, x) for all x, y ≥ 0. This completes the proof of Theorem 8.12.5. 8.12.12. Proof of Theorem 8.12.6. By Lemma 8.12.10 we may already assume that W (λ) ∈ GL(H) Let Φ(z) := z

for all λ ∈ R.

(8.12.22)

1+z . 1−z

As W (∞) = I, then it follows from (8.12.22) that (W ◦ Φ)(z) is invertible for all z ∈ T, where T is the unit circle. Moreover, part (ii) of Corollary 8.12.4 in particular implies that W ◦ Φ admits local factorizations with respect to T and GL(H) (Def. 7.1.3). Therefore, by Theorem 8.4.2, the Wiener-Hopf operator WW ◦Φ is a Fredholm operator, if and only if, W ◦ Φ admits a factorization relative to T and GL(H). As, by Proposition 8.12.9, WW ◦Φ = V−1 UTW U−1 V,

(8.12.23)

this implies that TW is a Fredholm operator, if and only if W ◦ Φ admits a factorization relative to T and GL(H). Finally, by Proposition 8.11.3, this yields that TW is a Fredholm operator, if and only if, W admits a canonical factorization relative to R and GL(H). So the equivalence of the conditions (i) and (ii) in Theorem 8.12.6 is proved.

334

Chapter 8. Wiener-Hopf operators, Toeplitz operators Now we assume that these two conditions are satisfied, that κ n λ−i j Pj W+ (λ), λ ∈ R, W (λ) = W− (λ) P0 + λ+i j=1

is a factorization of W relative to R and GL(H), and that r is the index with κ1 > . . . > κr > 0 > κr+1 > . . . > κn . By Proposition 8.11.3, then n (W ◦ Φ)(z) = (W− ◦ Φ)(z) P0 + z κj Pj (W+ ◦ Φ)(z), z ∈ T, j=1

is a factorization of W ◦Φ relative to T and GL(H). By Theorem 8.4.2, this implies that dim Ker WW ◦Φ = −

n

κj Pj

and

dim Coker WW ◦Φ =

j=r+1

r

κj Pj .

j=1

By (8.12.23), this further implies (8.12.10). This completes the proof of Theorem 8.12.6. 8.12.13 Proposition. Let V, W ∈ S R, L(H) . Then V, W ∈ S R, L(H) and if KV and KW are the kernel functions of V and W , respectively, then KV + KW − KV ∗ KW is the kernel function of V W . !W = K !V K Proof. As K V ∗ KW (see (8.7.12)), we have ! V )(I − K !W = I − K !W ) = I − K !V − K !W + K !V K !V − K ! W + KV V W = (I − K ∗ KW .

8.12.14 Proposition. Let W− ∈ S− S R, L(H) . Then

R, L(H) , W+ ∈ S+ R, L(H) and W ∈

(i) TW− W = TW− TW and TW W+ = TW TW+ . (ii) Suppose W− (λ) is invertible for all λ ∈ H− ∪ {∞}, and, hence (Corollaries 8.12.3 and 8.12.4), also W−−1 ∈ S− R, L(H) . Then TW− is invertible and −1 TW = TW −1 . −

−

(iii) Suppose W+ (λ) is invertible for all λ ∈ H+ ∪ {∞}, and, hence (Corollaries 8.12.3 and 8.12.4), also W+−1 ∈ S+ R, L(H) . Then TW+ is invertible and −1 TW = TW −1 . +

+


335

We only have to prove part (i), because (ii) and (iii) follow from (i). Let K± and K be the kernel functions of W± and W , respectively. Then, for all f ∈ L2+ (R, H),

TW− TW f = TW− f − P R (K ∗ f )

= f − P R (K ∗ f ) − P R K− ∗ f − P R (K ∗ f )

= f − P R (K + K− ) ∗ f + P R K− ∗ P R (K ∗ f ) and

TW TW+ f = TW f − P R (K+ ∗ f )

= f − P R (K+ ∗ f ) − P R K ∗ f − P R (K+ ∗ f )

= f − P R (K+ + K) ∗ f + P R K ∗ P R (K+ ∗ f ) .

Since, by proposition (8.8.8),

P R K− ∗ P R (K ∗ f ) = P R K− ∗ (K ∗ f )

P R K ∗ P R (K+ ∗ f ) = P R K ∗ (K+ ∗ f ) ,

and this implies that

TW− TW f = f − P R (K + K− ) ∗ f + P R K− ∗ (K ∗ f ) and

TW TW+ f = f − P R (K+ + K) ∗ f + P R K ∗ (K+ ∗ f ) .

As the convolution is associative (Proposition 8.8.7), this further implies

TW− TW f = f − P R (K + K− − K− ∗ K) ∗ f and

TW TW+ f = f − P R (K+ + K − K ∗ K+ ) ∗ f .

Since, by Proposition 8.12.13, K + K− − K− ∗ K is the kernel function of TW− W and K+ + K − K ∗ K+ is the kernel function of TW W+ , this proves part (i) of the proposition. 8.12.15. Proof of Theorem 8.12.6. We introduce the abbreviations Ψ(λ) =

λ−i λ+i

and

Θ(λ) =

λ+i , λ−i

λ ∈ C.

336


Then, by definition of Δ, r

Δ = P0 +

j=1

Δ

−1

= P0 +

n

Ψκj Pj +

r

κj

Θ−κj Pj ,

j=r+1 n

(8.12.24) −κj

Θ Pj +

j=1

Ψ

Pj ,

j=r+1

and, by (8.12.5), Ψκ ∈ S+ R, C

and Θκ ∈ S− R, C ,

κ ∈ N∗ .

By Proposition 8.12.14, the latter relation implies that κ ∈ N∗ .

TΘκ TΨκ = I,

(8.12.25)

From (8.12.24) it follows that r

TΔ = P0 +

T

Ψκj

Pj +

j=1

TΔ−1 = P0 +

r

r

TΘ−κj Pj ,

j=r+1 n

(8.12.26)

TΘκj Pj +

j=1

TΨ−κj Pj ,

j=r+1

and, further, by (8.12.25), −1 TΔ−1 TΔ−1 TΔ

= P0 +

r

T

Θκj

T

Ψκj

T

Θκj

Pj +

j=1

−1 TΔ TΔ −1 TΔ

r

TΨ−κj TΘ−κj TΨ−κj Pj

j=r+1

= TΔ−1 , r n = P0 + TΨκj TΘκj TΨκj Pj + TΘ−κj TΨ−κj TΘ−κj Pj j=1

j=r+1

= TΔ . Moreover, again by Proposition 8.12.14, TW −1 TW− = TW− TW −1 = I, −

−

TW −1 TW+ = TW+ TW −1 = I, +

T W = T W− T Δ T W + .

+


337

Together this yields

TW TW −1 TΔ−1 TW −1 TW + −

= TW− TΔ TW+ TW −1 TΔ−1 TW −1 TW− TΔ TW+ +

−

= TW −1 TΔ−1 TΔ TΔ−1 TW −1 −

+

= TW −1 TΔ−1 TW −1 −

+

= TW and

TW −1 TΔ−1 TW −1 TW TW −1 TΔ−1 TW −1 + − + −

= TW −1 TΔ−1 TW −1 TW− TΔ TW+ TW −1 TΔ−1 TW −1 +

−

+

−

= TW −1 TΔ−1 TΔ TΔ−1 TW −1 +

−

= TW −1 TΔ−1 TW −1 , +

−

which shows that TW −1 TΔ−1 TW −1 is a generalized inverse of TW . +

−

It remains to prove (8.12.12). It follows from Proposition 8.7.11 that (Ψ − 1)ν =

(−2)ν !+ θ ν! ν−1

and (Θ − 1)ν =

2ν !− , θ ν! ν−1

ν ∈ N∗ ,

and further κ

Ψκ − 1 = (Ψ − 1 + 1) − 1 = = Θ −1= κ

κ κ (−2)ν ν=1 κ ν=1

ν

ν!

κ κ ν=1

+ , θ!ν−1

κ 2ν !− , θ ν ν! ν−1

ν

(Ψ − 1)ν

κ ∈ N∗ , κ ∈ N∗ .

This implies that, for all u ∈ L2+ (R, H), κ κ (−2)ν ∞ !+ θν−1 (x)u(x)dx, y ≥ 0, TΨκ u (y) = u(y) + ν ν! −∞ ν=1 κ ν ∞ κ 2 − (x)u(x)dx, y ≥ 0. TΘκ u (y) = u(y) + θ!ν−1 ν ν! −∞ ν=1

338


± (Section 8.7.8), for all ν ∈ N∗ , Since, by definition of the functions θν−1 if y − x ≥ 0, (y − x)ν−1 ex−y + (y − x) = θν−1 0 if y − x < 0,

and

−(y − x)ν−1 ey−x − θν−1 (y − x) = 0

if y − x ≤ 0, if y − x > 0,

and since u(x) = 0 for x < 0 if u ∈ L2+ (R, C), this further implies that, for all ν ∈ N∗ , κ κ (−2)ν y (TΨκ u)(y) = u(y) + (y − x)ν−1 ex−y u(x)dx, y ≥ 0, ν ν! 0 ν=1 (8.12.27) κ ν ∞ κ 2 ν−1 y−x (y − x) e u(x)dx, y ≥ 0. TΘκ u (y) = u(y) − ν ν! y ν=1 Set

TΘκj Tj = TΨ−κj

for 1 ≤ j ≤ r, for r + 1 ≤ n.

Then (8.12.12) follows from (8.12.26) and (8.12.27).

8.13

An example

In this section, D+ ⊆ C is a bounded connected open set with piecewise C 1 boundary Γ such that 0 ∈ D+ . If E is a Banach space, and A ∈ L(E) is a Fredholm operator, then (see, e.g., [GGK2]) there exists ε > 0 such that the following holds: If B ∈ L(E) with

A − B < ε, then also B is a Fredholm operator, where ind A = ind B, and if A is invertible, then also B is invertible. In view of the connection with Wiener-Hopf operators, this implies different stability statements for the factorization problem. For example, Theorem 8.4.2 immediately implies: 8.13.1 Corollary (to Theorem 8.4.2). Let H be a separable Hilbert space, and let A : Γ → GL(H) be a continuous function which admits a factorization with respect to Γ (Def. 7.1.1). Then there exists ε > 0 such that the following holds: Let B : Γ → GL(H) be a continuous function which admits local factorizations with respect to Γ (Def. 7.1.3) and which satisfies the estimate max A(z) − B(z) L(H) ≤ ε . z∈Γ

Then also B admits a factorization with respect to Γ and, moreover:

8.13. An example

339

(i) If κ1 (A), . . . , κn (A) and κ1 (B), . . . , κm (B) are the non-zero partial indices of A and B, respectively (Def. 7.9.6), if dj (A) is the multiplicity of κj (A) as a partial index of A (Def. 7.9.8), and if dj (B) is the multiplicity of κj (B) as a partial index of B, then n j=1

κj (A)dj (A) =

m

κj (B)dj (B) .

(8.13.1)

j=1

(ii) If A admits a canonical factorization with respect to Γ, then also B admits a canonical factorization with respect to Γ. However, each partial index alone is not stable with respect to small pertubations. Consider, for ε ∈ C, the following example of a 2 × 2-matrix: −1 0 z Aε (z) := , z ∈ C∗ . (8.13.2) ε z For ε = 0, the partial indices with respect to Γ of this function are −1 and 1. However, for all ε = 0, Aε admits a canonical factorization with respect to Γ, namely: −1 −1 z 0 z −ε−1 1 ε−1 z = . (8.13.3) ε z 0 1 ε 0 This example is good to explain also some other questions. Factorizations with respect to Γ as considered in this book, are also called right factorizations with respect to Γ. The notion of a left factorization with respect to Γ one obtains by interchanging the roles of A− and A+ in Definition 7.1.1, i.e., a left factorization of a continuous function A : Γ → GL(E) (E being a Banach space) with respect to Γ is a representation of the form A = A+ ΔA− , where A− , A+ and Δ are as in Definition 7.1.1. We restrict ourselves to right factorizations with respect to Γ, because the theories of right and left factorizations with respect to Γ are equivalent. But we point out that the partial indices (Def. 7.9.6) need not be the same for the left and the right factorization with respect to Γ of the same function (if both exist). An example is given by the function Aε above if ε = 0. Then (8.13.3) shows that zero is the only “right” partial index of Aε with respect to Γ, whereas the representation −1 −1 z 0 1 0 z 0 = . (8.13.4) ε z εz 1 0 z shows that the “left” partial indices of Aε with respect to Γ are −1 and 1. Finally, we discuss the following generalization. Let E be a Banach space, and let A : Γ → GL(E) be a continuous function. A representation of the form A = A− ΔA+ will be called a generalized factorization of A with respect to Γ if everything is as in Definition 7.1.1, with one exception: We do not require that the projectors P1 , . . . , Pn are finite dimensional.

340


It is easy to see that in terms of the filtration of A with respect to Γ (Def. 7.9.8), this can be characterized as follows: A generalized factorization of A with respect to Γ exists, if and only if A admits local factorizations with respect to Γ and: If k1 > . . . > kn are the partial indices of A with respect to Γ (Def. 7.9.6), then, for 1 ≤ j ≤ n − 1, the spaces M− (z, kj , Γ, A), z ∈ D− ∪ {∞}, and M+ (z, kj , Γ, A), z ∈ D+ , are topologically closed and complemented in M− (z, kj+1 , Γ, A) and M+ (z, kj+1 , Γ, A), respectively, and the families of subspaces M− (z, kj , Γ, A) and M+ (z, kj , Γ, A) z∈D − ∪{∞}

z∈D +

are continuous (Def. 6.2.1) and holomorphic (Def. 6.4.1) over D− ∪ {∞} and D+ , respectively. It turns out that also a generalized factorization of A with respect to Γ does not always exist, even if A is a polynomial in z and z −1 . To show this, consider the following infinite dimensional version of example (8.13.3): Let H be an infinite dimensional separable Hilbert space and let V ∈ L(H) be an operator such that Ker V = {0} and Im V is not topologically closed in H, and let A : C∗ → L(H ⊕H) be defined by the block matrix −1 z I 0 A(z) = , z ∈ C∗ , (8.13.5) V zI where I is the unit operator on H. It is easy to see that the only (Γ, −1) section of A (Def. 7.9.1) is the zero section, and that the space of (Γ, 0)-sections of A consists of all pairs (ϕ− , ϕ+ ) of the form −1 a + zb z a+b with a, b ∈ H . ϕ+ (z) = and ϕ− (z) = Va −V b Hence, 0 is a partial index of A with respect to Γ, and M+ (0, 0, Γ, A) = H ⊕ Im V , which is not topologically closed in H ⊕ H (as Im V is not topologically closed in H). Therefore, A does not admit a generalized factorization with respect to Γ.

8.14

Comments

The material of this chapter for matrix-valued functions was first presented in [GK] (see also [Go4, CG]). It consists of an infinite dimensional generalization for a wider set of functions. Some operator generalization of the Wiener-Hopf equation one can find in the paper of Feldman [F]. The presented operator-valued generalizations are used for the solution of the linear transport equation. The transport theory concerns the mathematical analysis of equations that describe transport phenomena in matter, e.g. a flow of electrons through a metal strip or radiated

8.14. Comments

341

transport bin stellar atmosphere. This phenomenon concerns the migration of particles in a medium. For the finite dimensional case, see [BGK], chapter 6. For the linear transport theory, see [KLH].

Chapter 9

Multiplicative cocycles with restrictions (F-cocycles) Here we present a generalization of the theory of cocycles developed in Chapter 5. We study multiplicative cocycles with restrictions. To formulate this, it is convenient to use the language of sheaves.

9.1 F-cocycles In this section, A is a Banach algebra with unit 1, and G is an open subgroup of the group of invertible elements of A. By P1 we denote the Riemann sphere (see the beginning of Section 5.10). If U ⊆ P1 is a non-empty open set, then we denote by OG (U ) the group of G-valued holomorphic functions defined on U , and we set OG (∅) = {1}. 9.1.1 Definition (OG -sheaf ). Let D ⊆ P1 be an open set. A map F, which assigns to each open set U ⊆ D a subgroup F(U ) of OG (U ), will be called an OG -sheaf over D if, for each open U ⊆ D, the following two conditions are satisfied: (i) If f ∈ F(U ), then f V ∈ F(V ) for each open V ⊆ U . Here f |∅ := 1. (ii) Suppose f ∈ OG (U ) such that, for each w ∈ U there exists a neighborhood W ⊆ U of w with f W ∈ F(W ). Then f ∈ F(U ). If F is an OG -sheaf over D and U ⊆ D is open, then the functions from F(U ) are called sections of F over U . If F is an OG -sheaf over D and Y ⊆ D is open, then we denote by F Y the restriction of F to the open subsets of Y .

344

Chapter 9. Multiplicative cocycles with restrictions

9.1.2 Definition (Sheaves defined by a data of zeros). A pair (Z, m) is called a data of zeros if Z ⊆ P1 and m = {mw }w∈Z is a family of numbers mw ∈ N. Let (Z, m) be a data of zeros. Then, for each open set U ⊆ P1 , we denote by G OZ,m (U ) the group of all f ∈ OG (U ) such that, for each w ∈ U ∩ Z, the function G f − 1 has a zero of order ≥ mw at w.1 The map OZ,m is an OG -sheaf over P1 . The G restriction of OZ,m to the open subsets of an open set D ⊆ P1 will be denoted by G OD,Z,m . 9.1.3 Definition (OG -sheaves of finite order). Let D ⊆ P1 be an open set. An OG -sheaf F over D will be called of finite order if there exists a data of zeros (Z, m) such that Z ∩ D is discrete and closed in D and, for each open set U ⊆ D, F(U \ Z) = OG (U \ Z) ,

(9.1.1)

G F(U ) ⊇ OZ,m (U ) .

(9.1.2)

In particular, if (Z, m) is a data of zeros and D ⊆ P1 is an open set such that G Z ∩ D is discrete and closed in D, then OD,Z,m is of finite order. 9.1.4 Definition ((U, F)-cocycles). Let D ⊆ P1 be an open set, let F be an OG sheaf over D, and let U = {Uj }j∈I be an open covering of D. Then we denote by Z 1 (U, F) the set of all f ∈ Z 1 (U, OG ) (Def. 5.6.1) with fjk ∈ F(Uj ∩Uk ), j, k ∈ I. The elements of Z 1 (U, F) will be called (U, F)-cocycles. Two cocycles f, g ∈ Z 1 (U, F) will be called F-equivalent if there exists a family h = {hj }j∈I of functions hj ∈ F(Uj ) such that hj fjk h−1 k = gjk

on Uj ∩ Uk

(9.1.3)

for all j, k ∈ I with Uj ∩ Uk = ∅. A cocycle f ∈ Z 1 (U, F) will be called F-trivial if it is equivalent to the cocycle {ejk }jk∈I defined by ejk ≡ 1 on Uj ∩ Uk for all j, k ∈ I with Uj ∩ Uk = ∅. More directly, a cocycle f ∈ Z 1 (U, F) will be called F-trivial if there exists a family {hj }j∈I of sections fj ∈ F(Uj ) such that fjk = h−1 j hk

on Uj ∩ Uk

(9.1.4)

for all j, k ∈ I with Uj ∩ Uk = ∅. The family {hj }j∈I then will be called a splitting of f . 9.1.5 Definition (Passing to refinements). Let F be an OG -sheaf over an open set D ⊆ P1 , and let U = {Uj }j∈I and V = {Vj }j∈J be two open coverings of D such that V is a refinement of U. If τ : J → I is a map with Vj ⊆ Uτ (j) for all j ∈ J (by definition of a refinement at least one such map exists), then, for each f ∈ Z 1 (U, F), we define −1 n f −1 = 1 − (1 − f ) = 1 + (1 − f ) ∞ n=0 (1 − f ) in some neighborhood of w if f − 1 has a zero at w, this is indeed a group. 1 Since

9.1. F-cocycles

345

a cocycle τ ∗ f ∈ Z 1 (V, F), by setting (τ ∗ f )jk = fτ (j)τ (k) V

j ∩Vk

,

j, k ∈ J .

Here again fτ (j)τ (k) ∅ := 1. We shall say that g ∈ Z 1 (V, F) is induced by f ∈ Z 1 (U, F) if there exists a map τ : J → I with Vj ⊆ Uτ (j) , j ∈ J, and g = τ ∗ f . There is the following generalization of Proposition 5.7.2: 9.1.6 Proposition. Let F be an OG -sheaf over an open set D ⊆ P1 , and let U = {Uj }j∈I and V = {Vj }j∈J be two open coverings of D such that V is a refinement of U. If f, g ∈ Z 1 (U, F) and f, g ∈ Z 1 (V, F) such that f is induced by f and g is induced by g, then the following are equivalent: (i) f and g are F-equivalent. (ii) f and g are F-equivalent. In particular, the following are equivalent: (i ) f is F-trivial. (ii ) f is F-trivial. Proof. Repetition of the proof of Proposition 5.7.2.

9.1.7 Definition (F-cocycles). Let D ⊆ P be an open set, and let F be an O sheaf over D. 1

G

(i) By an F-cocycle we mean a (U, F)-cocycle such that U is an open covering of D. The covering U then is called the covering of this cocycle. (ii) Let f and g be two F-cocycles over D (possibly with different coverings). The cocycles f and g will be called F-equivalent if the following two equivalent (by Proposition 9.1.6) conditions are satisfied: 1) There exists an open covering W of D, which is a refinement both of the covering of f and of the covering of g, such that at least one of the (W, F)-cocycles induced by f is F-equivalent to at least one of the (W, F)-cocycles induced by g. 2) For each open covering W of D, which is a refinement both of the covering of f and of the covering of g, each (W, F)-cocycle induced by f is F-equivalent to each (W, F)-cocycle induced by g. 9.1.8 Definition (Restriction to subsets). Let D ⊆ C be an open set, let U = {Uj }j∈I be an open covering of D, and let Y be an open subset of D. Set U ∩ Y = Uj ∩ Y j ∈ I .

346


Let F be an OG -sheaf over D. Then we define: (i) Let f be an F-cocycle over D with the covering U. Then we denote by f Y the F Y -cocycle with the covering U ∩ Y defined by (f |Y )jk = fjk U ∩U ∩Y j

k

for j, k ∈ I with Uj ∩ Uk ∩ Y = ∅. This cocycle f Y will be called the restriction of f to Y . We shall say that f is F-trivial over Y if f Y is F-trivial. (ii) Let f, g be two F-cocycles over D. Then we shall say that f and g are F-equivalent over Y if f Y and g Y are F Y -equivalent. 9.1.9 Proposition. Let D ⊆ C be an open set, let U = {Uj }j∈I be an open covering of D, let F be an OG -sheaf over D, and let f be an F-cocycle over D, which is F-trivial over each Uj . Then f is F-equivalent to some (U, F)-cocycle. Proof. Repetition of the proof of Proposition 5.7.6

9.2

The main results on cocycles with restrictions. Formulation and reduction to OD,Z,m

In this section we use the notations and definitions introduced in the preceding section. The main results for cocycles obtained in this chapter can be stated as follows: 9.2.1 Theorem. Let F be an OG -sheaf of finite order over an open set D ⊆ C, and let f be an F-cocycle. Assume that at least one of the following three conditions is satisfied: (i) The cocycle f is C G -trivial over D (Definition 5.6.1). (ii) The group G is connected. (iii) D is simply connected. Then f is F-trivial. Note also the following corollary: 9.2.2 Corollary. Let F be an OG -sheaf of finite order over an open set D ⊆ C, and let D1 , D2 ⊆ D be two open sets with D = D1 ∪ D2 . Further let f ∈ F(D1 ∩ D2 ) and assume that at least one of the following three conditions is satisfied: (i) There exist continuous functions cj : Dj → G, j = 1, 2, such that f = c−1 1 c2 on D1 ∩ D2 . (ii) The group G is connected. (iii) D1 ∪ D2 is simply connected.

9.3. The Cartan lemma with restrictions

347

Then there exist functions fj ∈ F(Dj ), j = 1, 2, such that f = f1−1 f2 on D1 ∩ D2 . 9.2.3 Remark. It is sufficient to prove Theorem 9.2.1 for the case when F is of G the form OD,Z,m , where (Z, m) is a data of zeros such that Z ∩ D is discrete and closed in D (Def. 9.1.2). Indeed, assume this is done, and let F be an arbitrary OG -sheaf of finite order over an open set D ⊆ C. Further, let f be an F-cocycle such that at least one of the following three conditions (i)–(iii) in Theorem 9.2.1 is satisfied. Let U be the open covering of D associated to f , and let (Z, m) be a data of zeros as in Definition 9.1.3. Since Z ∩ D is discrete and closed in D, then, for each point w ∈ D, we can find a neighborhood Vw ⊆ D of w such that Vw is contained in at least one of the sets of the covering U and {w} if w ∈ Z , Vw ∩ Z = ∅ if w ∈ Z . Then V := {Vw }w∈D is an open covering of D and a refinement of U. Let f ∗ be a (V, F)-cocycle induced by f . If w, z are two different points in D, then Z∩Vw ∩Vz = ∅. By (9.1.1) this implies that G F(Vw ∩ Vz ) = OG (Vw ∩ Vz ) = OZ,m (Vw ∩ Vz )

if w, z ∈ D with w = z .

G Therefore f ∗ can be interpreted as a (V, OD,Z,m )-cocycle. If one of the conditions (ii) or (iii) is satisfied for F, then it is clear that this G condition is satisfied for OD,Z,m (as it depends only on D resp. G). If condition (i) is satisfied for the cocycle f , then it follows from Proposition 9.1.6 that this condition is satisfied also for f ∗ . G Therefore, as Theorem 9.2.1 is already proved for the sheaf OD,Z,m (by our ∗ G hypothesis), it follows that f is OD,Z,m -trivial. By (9.1.2) this implies that f ∗ is F-trivial, which means by Proposition 9.1.6 that f is F-trivial.

The following Sections 9.3–9.8 are devoted to the proof of Theorem 9.2.1 in G the case when F is of the form OD,Z,m , where (Z, m) is a data of zeros such that Z ∩D is discrete and closed in D. Then the assertion of Theorem 9.2.1 is contained in Theorems 9.6.1 (if D is simply connected) and 9.8.1 (if f is C G -trivial or G is connected).

9.3

The Cartan lemma with restrictions

In this section, A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, G1 A is the connected component of 1 in G, and (Z, m) is a data of zeros (Def. 9.1.2). G Here we prove a version of the Cartan lemma for sections of the sheaf OZ,m . To linearize the problem, we need the following definition:

348


A (U ) the set of all 9.3.1 Definition. For each open set U ⊆ P1 , we denote by OZ,m A f ∈ O (U ) such that, for each point w ∈ U ∩ Z, the function f has a zero of order ≥ mw at w.

9.3.2 Lemma. (For the definitions of log and exp, see Section 5.4.3). For each open set U ⊆ P1 , we have: G1 A A (i) If f ∈ OZ,m (U ), then exp f ∈ OZ,m (U ). G1 A A (U ) and f (z) − 1 < 1, z ∈ U , then log f ∈ OZ,m (U ). (ii) If f ∈ OZ,m A (U ) and w ∈ Z. Then Proof. (i) Let f ∈ OZ,m

exp f − 1 =

∞ fn . n! n=1

As f has a zero of order ≥ mw at w and therefore each f n with n ≥ 1 has a zero of order ≥ mw at w, this implies that exp f − 1 has a zero of order ≥ mw at w. G1 A (U ) with f (z) − 1 < 1 for all z ∈ U , and let w ∈ Z. Then (ii) Let f ∈ OZ,m

log f =

∞ (−1)n+1 (f − 1)n . n n=1

As f − 1 has a zero of order ≥ mw at w and therefore each (f − 1)n with n ≥ 1 has a zero of order ≥ mw at w, this implies that log f has a zero of order ≥ mw at w. 9.3.3 Definition. Let Γ be an arbitrary subset of P1 , and let Γ0 be the set of interior points of Γ. G

(i) We denote by OZ,m (Γ) the group of continuous functions f : Γ → G such G

G that f |Γ0 ∈ OZ,m (Γ0 ).2 If Γ is compact, then OZ,m (Γ) will be considered as a topological group with respect to uniform convergence on Γ. It is easy to see that this is a closed subgroup of C G (Γ) endowed with the same topology. A

(ii) We denote by OZ,m (Γ) the algebra of continuous functions f : Γ → A such A

A that f |Γ0 ∈ OZ,m (Γ0 ). If Γ is compact, then OZ,m (Γ) will be considered as the Banach algebra endowed with the max-norm. It is easy to see that this is indeed a Banach algebra, i.e., a closed subalgebra of the Banach algebra C A (Γ) endowed with the same norm. 2 Here

G

it is allowed that points of Z lie on Γ \ Γ0 , but for f ∈ OZ,m (Γ), the function f − 1 need not have zeros at such points.


349

9.3.4 Proposition. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, and let Ω ⊇ D be an open set such that each bounded connected component of C\D contains at least one point of C \ Ω. Assume that Z ∩ Ω is discrete and closed in Ω and Z ∩ ∂D = ∅. A Then each f ∈ OZ,m (D) can be approximated uniformly on D by functions A from OZ,m (Ω). Proof. Let U1 , . . . , Un be the bounded connected components of C \ D. By hypothesis, we can find points aj ∈ Uj ∩ (C \ Ω), 1 ≤ j ≤ n. Since Z ∩ Ω is discrete and closed in Ω, by the Weierstrass product Theorem 2.5.7, we can find a scalar holomorphic function ϕ : Ω → C such that ϕ(z) = 0 for z ∈ Ω \ Z, and, for each w ∈ Z, ϕ has a zero of order mw at w. Since Z ∩ ∂D = ∅, then f f := ϕ is continuous on D and holomorphic in D. Therefore, by the Runge approximation Theorem 2.2.2 (ii), we can find a sequence (fν )ν∈N of functions fν ∈ OA C \ {a1 , . . . , an } which converges to f uniformly on D. Setting fν = ϕfν on Ω, we get A a sequence (fν )ν∈N of functions fν ∈ OZ,m (Ω). Since D is compact and, therefore, ϕ is bounded on D, this sequence converges to f = ϕf uniformly on D. 9.3.5 Lemma. Let Ω ⊆ C be a bounded open set with piecewise C 1 -boundary such that C \ Ω is connected. Suppose that Z ∩ C is discrete and closed in C and Z ∩ Ω = ∅.

3

Then each f ∈ OG1 A (Ω) can be approximated uniformly on Ω by maps from G1 A OZ,m (C). Proof. Let f ∈ OG1 A (Ω) be given. By Lemma 5.4.7, OG1 A (Ω) is the connected component of the unit in the group of invertible elements of the Banach algebra OA (Ω). Therefore, by Proposition 5.4.1, f can be written as a finite product f = g1 · . . . · gn where gj ∈ OG1 A (Ω) and 1 − gj OA (Ω) < 1 for 1 ≤ j ≤ n. By Lemma 5.4.4 (ii), elog gj = gj . Hence f = elog g1 · . . . · elog gn . Since Z ∩ Ω = ∅, it is trivial that log gj ∈ OA Z,m (Ω). Since C \ Ω is connected, by Proposition 9.3.4 (i), for each j, we can find a sequence (ϕjν )ν∈N of functions A ϕjν ∈ OZ,m (C) which converges to log gj uniformly on Ω. Setting fν = eϕjν . . . eϕjν 3 We

shall see below (Theorem 9.5.3) that this condition can be replaced by Z ∩ ∂Ω = ∅.

350


then we get a sequence (fν )ν∈N of holomorphic functions fν : C → A which G1 A converges to f uniformly on D. By Lemma 9.3.2 (i) each fν belongs to OZ,m (C). A modification of the proof of Lemma 5.5.3 gives the following generalization of this lemma: 9.3.6 Lemma. Let (D1 , D2 ) be a Cartan pair such that Z ∩ ∂D1 = Z ∩ ∂D2 = ∅

Z ∩ (D1 ∪ D2 ) is finite.

and

A

Let ε > 0. Then there exists δ > 0 such that, for all g ∈ OZ,m (D1 ∩ D2 ) with max

z∈D 1 ∩D 2

g(z) < δ,

A

there exist gj ∈ OZ,m (Dj ) such that max gj (z) < ε, j = 1, 2, and z∈D j

1 + g = (1 + g1 )(1 + g2 )

on D1 ∩ D2 .

(9.3.1)

A

A Proof. We consider OZ,m (D1 ∩ D2 ), OA Z,m (D 1 ) and O Z,m (D 2 ) as Banach algebras A

endowed with the maximum norm. It is sufficient to prove that each f ∈ OZ,m (D1 ∩ D2 ) can be written in the form f = f1 + f2 with fj ∈ OA Z,m (D j ), because then the assertion follows from Lemma 5.2.1. If Z ∩ (D1 ∪ D2 ) = ∅, this follows from Corollary 5.3.5. If Z ∩ (D1 ∪ D2 ) = {w1 , . . . , wn }, n ∈ N∗ , then we set p(z) = (z − z1 )mw1 · . . . · (z − zn )mwn . A

Now let an arbitrary f ∈ OZ,m (D1 ∩ D2 ) be given. Since Z does not meet the A

boundaries of D1 and D2 , then f /p ∈ O (D1 ∩ D2 ), and, again by Corollary 5.3.5, A we can find functions hj ∈ O (Dj ) with f /p = h1 + h2 . Setting fj = phj , we get A

the required functions fj ∈ OZ,m (Dj ) with f = f1 + f2 .

Now we can prove the following generalization of the Cartan Lemma 5.5.2: 9.3.7 Lemma. Let (D1 , D2 ) be a Cartan pair such that Z ∩ ∂(D1 ∪ D2 ) = ∅

and

Z ∩ (D1 ∪ D2 ) is finite.

OG Z,m (D 1

∩ D2 ) such that all values of f belong to the same connected Let f ∈ component of G. (This is always the case if D1 ∩ D2 is connected.) Then there exist fj ∈ OG Z,m (D j ), j = 1, 2, such that f = f1 f2 4 From

on

D1 ∩ D2 .

4

(9.3.2)

the proof of this lemma it will be clear that also the more precise statements corresponding to assertions (i) and (ii) of Lemma 5.5.2 are true. We omit these finer points, because we will not use them.


351

Proof. Since all values of f belong to the same connected component of G, after multiplying by a constant, we may assume that all values of f belong to G1 A. The main problem compared to the proof of Lemma 5.5.2 is that, possibly, Z ∩ D1 ∩ D2 = ∅. We avoid this problem by the following trick: Since Z ∩ (D1 ∪ D2 ) is finite, we can find a ”smaller” Cartan pair (X1 , X2 ) such that Xj ⊆ Dj , X1 ∪ X2 = D1 ∪ D2 and Z ∩ X 1 ∩ X 2 = ∅. G1 A (X j ), j = 1, 2, such that Now it is sufficient to find fj ∈ OZ,m

f = f1 f2

on X 1 ∩ X 2 .

(9.3.3)

Indeed, then it follows from the equations f1 = f f2−1

and

f2 = f1−1 f

1A that the maps fj extend to maps fj ∈ OG Z,m (D j ), and (9.3.2) follows from (9.3.3) by uniqueness of holomorphic functions. Now we continue as in the proof of Lemma 5.5.2: Since Z ∩ X 1 ∩ X 2 = ∅, it follows from Lemma 9.3.5 that, for each δ > 0, we G1 A can find fδ ∈ OZ,m (C) with

max

z∈U 1 ∪...∪U n

f (z)fδ−1 (z) − 1 < δ

and

max

z∈U 1 ∪...∪U n

fδ−1 (z)f (z) − 1 < δ.

Therefore, by the preceding Lemma 9.3.6, for sufficiently small δ, we can find gj ∈ OA (Dj ) such that max gj (z) < 1 z∈D j

and

f fδ−1 = (1 + g1 ) (1 + g2 )

on D1 ∩ D2 . Setting f1 = 1 + g1 and f2 = (1 + g2 )fδ , we conclude the proof.

9.3.8 Corollary. Let D ⊆ C be an open set such that Z ∩ D is discrete and closed in D. Let (D1 , D2 ) be a Cartan pair such that D1 ∪ D2 ⊆ D. G (U ) such that all values Let U ⊆ D be a neighborhood of D1 ∩ D2 and f ∈ OZ,m of f belong to the same connected component of G. (This is always the case if U is connected.) Then there exist neighborhoods Uj ⊆ D of Dj and functions G fj ∈ OZ,m (Uj ), j = 1, 2, such that

f = f1 f2

on

U1 ∩ U2 .

(9.3.4)

352


Proof. Since Z ∩ D is discrete and closed in D, we can find a slightly larger Cartan pair (U1 , U2 ) such that Dj ⊆ Uj , U j ⊆ D, U 1 ∩ U 2 ⊆ U and Z ∩ ∂(U1 ∪ U2 ) = ∅. Then Z ∩ (U1 ∪ U1 ) is finite, f |U 1 ∩U 2 ∈ OG Z,m (U 1 ∩ U 2 ) and we can apply Lemma 9.3.7 to the Cartan pair (U1 , U2 ).

9.4

Splitting over simply connected open sets after shrinking

In this section A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, and (Z, m) is a data of zeros (Def. 9.1.2). By G1 A we denote again the connected component of 1 in the group of all invertible elements of A. G Here we want to generalize the splitting Theorem 5.6.3 to OZ,m -cocycles. G The problem is that (at the moment) we do not have a generalization to OZ,m of part (i) of the Runge approximation Theorem 5.0.1, which we used in the proof of Lemma 5.9.1 and then again in the proof of Theorem 5.6.3. In the next section we will prove this generalization, but first we have to prove the following splitting statement “with shrinking”: 9.4.1 Lemma. Let D ⊆ C be a simply connected open set such that Z ∩D is discrete and closed in D, let Ω be a relatively compact open subset of D, and let f be an G G OZ,m -cocycle over D. Then the restriction f |Ω is OZ,m -trivial (Def. 9.1.4). Proof. We proceed similar to the proof of Lemma 5.9.2. But instead of the precise factorization statement of the Cartan Lemma 5.9.1, here we can use only the factorization statement “with shrinking” of Corollary 9.3.8. Therefore, in each of the induction steps, we have to shrink the covering which makes the arguments more technical. We give now the details. Set Kt = z ∈ C − t < Re z < 1 + t and − t < Im z < 1 + t , t > 0. Since D is simply connected and Ω is relatively compact in D, by the Riemann mapping theorem, we may assume that, for some ε > 0, D = Kε and Ω ⊆ K0 . G -cocycle f over D = Kε be given. Now let an OZ,m We choose n ∈ N∗ sufficiently large (will be specified some lines below) and t denote by Ujk , j, k = 1, . . . , n, 0 < t < 1, the open rectangle of all z ∈ C with k−1−

t 3

1 < Re z < n

k+

t 3

1 n

9.4. Splitting over simply connected open sets

and

t j−1− 3

1 < Im z < n

353

t j+ 3

1 . n

t }1≤j,k≤n is an open covering of Kt/3n . We Then, for all 0 < t < 1, U t := {Ujk choose n so large that

Kt/3n ⊆ Kε/2

for all 0 < t < 1.

Let V = {Vν }ν∈I be the covering associated to f . Then V ∩ Kt/3n := Vν ∩ Kt/3n

ν∈I

is a second open covering of Kt/3n . Since K ε/2 is a compact subset of D = Kε , the covering V contains a finite subcovering which already covers K ε/2 ⊇ Kt/3n . t Therefore, we can n further enlarge (and now we fix it) so that each Ujk is contained in at least one of the sets Vν . Now, for all 0 < t < 1, the covering U t is a refinement of V ∩ Kt/3n . t We give the family U t an order saying that Ujk < Ujt k , if and only if, either j < j or j = j and k < k . Let U1t , . . . Unt 2 be the family U t numbered in this way. For μ = 1, . . . , n2 , we consider the statement S(μ) : There exists 0 < t < 1 such that f U t ∪...∪U t 1

G is OZ,m -trivial.

μ

Since Ω ⊆ K0 ⊆ Kt = U1t ∪ . . . ∪ Unt 2 for all 0 < t < 1, it is sufficient to prove S(n2 ).Since U t is a refinement of V ∩ Kt/3n and V is associated to f , it is trivial G that f U t is OZ,m -trivial for all 1 ≤ μ ≤ n2 and 0 < t < 1. In particular, S(1) is μ true. Therefore it is sufficient to prove that S(μ) ⇒ S(μ + 1) for all 1 ≤ μ ≤ n2 − 1. Let 1 ≤ μ ≤ n2 − 1 be given such that S(μ) is true. Set W1t = U1t ∪ . . . ∪ Uμt t t and fix 0 < t < 1 such that W2 =GUμ+1 for 0 < t < 1. Since S(μ) is true, we can G f W t is OW t ,Z,m -trivial. Since, trivially, also f W t is OZ,m -trivial, it follows from 1 1 2 Proposition 9.1.9 that f t t is OG -equivalent to some {W1t , W2t }, OG W1 ∪W2

Z,m

Z,m

cocycle g. Now we choose some 0 < t < t. Then (W1t , W2t ) is a Cartan pair with t

t

t

t

W 1 ∪W 2 ⊆ D and W1t ∩W2t is a connected neighborhood of W 1 ∪W 2 . Therefore, from Corollary 9.3.8 we get functions gj ∈ OG Z,m (Wj ), j = 1, 2, such that g12 = g1 g1

on W1 ∩ W2 ,

G -trivial. Since W1 ∪ W2 = which means that the restricted cocycle g|W1 ∪W2 is OZ,m t , it follows that S(μ + 1) is true. U1t ∪ . . . ∪ Uμ+1

354

9.5


Runge approximation on simply connected open sets

In this section A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, G1 A is the connected component of 1 in the group of invertible elements of A, and (Z, m) is a data of zeros (Def. 9.1.2). Here we prove part (i) of the Runge approximation Theorem 5.0.1 for sections of the G sheaf OZ,m . The first step is the following Mergelyan approximation theorem: 9.5.1 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary such that Z ∩ D is finite and Z ∩ ∂D = ∅. Then, for each f ∈ OG Z,m (D) and all ε > 0, G there exists a neighborhood U of D and a function f ∈ OZ,m (U ) such that max f (z) − f(z) < ε. z∈D

Proof. Since the assertion of the theorem does not change if we replace Z by Z \ (C \ D), we may assume that Z is a finite subset of D. Then we can find an open set Ω ⊆ D with piecewise C 1 -boundary such that Ω ⊆ D and Z ⊆ Ω. (For example, we can surround the points of Z by small discs.) Further, let be an open disc such that D ⊆ . Now let f ∈ OG Z,m (D) and ε > 0 be given. Since G is an open subset of A, by the Mergelyan approximation Theorem 2.2.1, we can find a sequence (Un )n∈N of neighborhoods Un ⊆ of D and a sequence (hn )n∈N of functions hn ∈ OG (Un ) such that lim max f (z)h−1 n (z) − 1 = 0. n→∞ z∈D

Then, in particular lim

max

n→∞ z∈D∩(\Ω)

f (z)h−1 n (z) − 1 = 0.

(1) Therefore, by Lemma 9.3.6, we can find n0 ∈ N, a sequence gn n≥n of functions 0 (2) (1) (2) A (D) and a sequence g of functions g ∈ O gn ∈ OA n n Z,m Z,m ( \ Ω) = n≥n0 OG ( \ Ω) such that (1) f h−1 1 + gn(2) n = 1 + gn

and

on D ∩ ( \ Ω) , n ≥ n0 ,

lim max gn(1) (z) = 0

n→∞ z∈D

and

lim max gn(2) (z) = 0.

n→∞ z∈\Ω

(9.5.1)

(9.5.2)

9.5. Runge approximation on simply connected open sets

355

Moreover we can assume that 1 + gn(1) ∈ OG Z,m (D) Set

and

1 + gn(2) ∈ OG ( \ Ω) ,

−1 f fn := 1 + gn(1)

Then it follows from (9.5.1) that fn := 1 + gn(2) hn

n ≥ n0 .

on D.

on D ∩ ( \ Ω) , n ≥ n0 ,

G (Un ) for all n ≥ n0 , and from (9.5.2) it follows that which shows that fn ∈ OZ,m

lim max f (z) − fn (z) = 0.

n→∞ z∈D

It remains to choose n1 ≥ n0 so large that max f (z) − fn1 (z) < ε z∈D

and to set U = Un1 and f = fn1 .

G : Now we can prove Lemma 5.4.7 for the sheaf OZ,m

9.5.2 Lemma. Let D ⊆ C be a bounded simply connected open set with piecewise C 1 G1 A boundary. Suppose that Z ∩ D is finite and Z ∩ ∂D = ∅. Then the group OZ,m (D) is connected. G1 A (D) be given. Since G1 A is open and by the Mergelyan Proof. Let f ∈ OZ,m Theorem 9.5.1, we may assume that f is defined and holomorphic, and with values in G1 A in some neighborhood Ω of D. Since D is a bounded simply connected open set with piecewise C 1 -boundary, we may assume that also Ω is simply connected. Choose an open covering U = {Uj }j∈I of Ω by open discs Uj with center zj such that zj = zk if j = k and, for each j ∈ I, either Uj ∩ Z = ∅ or Uj ∩ Z = {zj }. Then Uj ∩ Uk ∩ Z = ∅ for j = k. Since G1 A is connected, for each j ∈ I, we can find a continuous curve γj : [−1, 0] → G1 A with γ(−1) = 1 and γ(0) = f (zj ). Moreover, if Uj ∩ Z = {zj } and hence f (zj ) = 1, then we may assume that γj ≡ 1. Now, for all j ∈ I and z ∈ Uj , we define γ(t) if −1 ≤ t ≤ 0 , φj (z, t) = f zj + t(z − zj ) if 0 ≤ t ≤ 1 .

So we get continuous G1 A-valued functions on Uj × [−1, 1], holomorphic for fixed t, such that φj (z, −1) = 1

and φj (z, 1) = f (z) ,

z ∈ Uj .

356


Moreover G1 A (Uj ) φj (·, t) ∈ OZ,m

for each fixed t ∈ [−1, 1] and j ∈ I.

(9.5.3)

Indeed, if Uj ∩ Z = ∅, this is trivial. If Uj ∩ Z = {zj } and −1 ≤ t ≤ 0, this is also trivial (because then γj ≡ 1). Now let Uj ∩ Z = {zj } and 0 ≤ t ≤ 1. Then f is of the form ∞ f (z) = 1 + fjν (z − zj )ν , z ∈ Uj , ν=mzj

and therefore φj (·, t) is of the form φj (z, t) = 1 +

∞

fjν tν (z − zj )ν ,

z ∈ Uj .

ν=mzj

This implies (9.5.3). Set ψjk (z, t) = φ−1 j (z, t)φk (z, t)

for z ∈ Uj ∩ Uk ,

j, k ∈ I .

(9.5.4)

be the algebra of all continuous maps ϕ : [−1, 1] → A such that, Now let A for some λ ∈ C, ϕ(−1) = ϕ(1) = λ · 1 . Endowed with the maximum norm, this is a Banach algebra with unit. Let GA be the group of invertible elements of A. It consists of all continuous functions ϕ : [−1, 1] → G1 A such that, for some λ ∈ C \ {0}, ϕ(−1) = ϕ(1) = λ · 1 .

(9.5.5)

With this notation, ψ := {ψjk (z, t)}j,k∈I can be considered as a (U, OGA )cocycle. Moreover, since Uj ∩ Uk ∩ Z = ∅ if j = k, it can be considered as a GA )-cocycle. (U, OZ,m Let W be a neighborhood of D with W ⊆ Ω. Since Ω is simply connected, GA -trivial. Hence we have functions then it follows from Lemma 9.4.1 that ψ|W is OZ,m

GA (W ∩ Uj ) such that, if we interpret them as functions ψj : W ∩ Uj × ψj ∈ OZ,m [−1, 1] → G1 A,

ψjk (z, t) = ψj−1 (z, t)ψk (z, t) ,

z ∈ W ∩ Uj ∩ Uk , j, k ∈ I .

Together with (9.5.4) this implies that φj (z, t)ψj−1 (z, t) = φk (z, t)ψk−1 (z, t) ,

z ∈ W ∩ Uj ∩ Uk , j, k ∈ I .

Therefore, we have a global continuous function F (z, t) : W × [−1, 1] → G1 A holomorphic for fixed t, such that, for each j ∈ I, F (z, t) = φj (z, t)ψj−1 (z, t) ,

z ∈ Uj , t ∈ [−1, 1].

(9.5.6)

9.5. Runge approximation on simply connected open sets

357

GA (W ∩ Uj ), implies that That ψj belongs to OZ,m G1 A (W ∩ Uj ) ψj (·, t) ∈ OZ,m

for all t ∈ [−1, 1] .

Together with (9.5.3) this implies that G1 A φj (·, t)ψ −1 (·, t) ∈ OZ,m (W ∩ Uj )

for all t ∈ [−1, 1] .

Hence, by (9.5.6), G1 A F (·, t) ∈ OZ,m (W )

Therefore

for all t ∈ [−1, 1] .

[−1, 1] t −→ F (·, t)D

(9.5.7)

G1 A

is a continuous curve in OZ,m (D).

GA (W ∩ Let us look for the beginning and the end of this curve. Since ψj ∈ OZ,m Uj ) and by (9.5.5), we have non-vanishing holomorphic functions λj : W ∩ Uj → C with ψj (z, −1) = ψj (z, 1) = λj (z) · 1 , z ∈ W ∩ Uj , j ∈ I .

Since φj (z, −1) = 1 and φj (z, 1) = f (z), this shows that

and

φj (z, −1)ψj−1 (z, −1) = λ−1 j (z) · 1 ,

z ∈ W ∩ Uj ,

j ∈I,

φj (z, 1)ψj−1 (z, −1) = λ−1 j (z)f (z) ,

z ∈ W ∩ Uj ,

j ∈I.

Together with (9.5.6) this gives F (z, −1) = λ−1 j (z)

and F (z, 1) = λ−1 j (z)f (z) ,

z ∈ W ∩ Uj ,

j ∈I.

This implies that there is a non-vanishing scalar holomorphic function λ globally defined on W with F (z, −1) = λ−1 (z) · 1

and F (z, 1) = λ−1 (z)f (z) ,

z∈W,

and the curve (9.5.7) connects the functions λ−1 f and λ−1 · 1. Hence [−1, 1] t −→ λF (·, t)D G1 A

is a continuous curve in OZ,m (D) which connects f and the unit element of G1 A

OZ,m (D).

9.5.3 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary such that C \ D is connected. Assume Z ∩ C is discrete and closed in C and Z ∩ ∂D = ∅. Let f ∈ OG Z,m (D) such that all values of f belong to the same connected component of G (which is always the case if D is connected). Then f can be approximated G uniformly on D by functions from OZ,m (C).

358


Proof. Since all values of f belong to the same connected component of G, after multiplying by a constant element of G, we may assume that f belongs to G1 A 1A OG Z,m (D). By Lemma 9.5.2, O Z,m (D) is connected. Therefore, by Proposition 5.4.1, f can be written in the form f = g1 · . . . · gn 1A where gj ∈ OG Z,m (D) and

max 1 − gj (z) < 1 z∈D

for 1 ≤ j ≤ n.

By Lemma 5.4.4 (ii), elog gj = gj . Hence f = elog g1 · . . . · elog gn . By Proposition 9.3.2, log gj ∈ OA Z,m (D). Therefore and since C\D is connected, we can apply Proposition 9.3.4 and get, for each j, a sequence (ϕjν )ν∈N of functions A ϕjν ∈ OZ,m (C) which converges to log gj uniformly on D. Setting fν = eϕjν · . . . · eϕjν then we get a sequence (fν )ν∈N of holomorphic functions fν : C → A which G1 A converges to f uniformly on D. By Proposition 9.3.2, each fν belongs to OZ,m (C).

9.6

Splitting over simply connected open sets without shrinking

In this section A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, G1 A is the connected component of 1 in the group of invertible elements of A, and (Z, m) is a data of zeros (Def. 9.1.2). It is the aim of this section to prove the following theorem: 9.6.1 Theorem. Let D ⊆ C be a simply connected open set such that Z ∩ D is G G discrete and closed in D. Then any OZ,m -cocycle over D is OZ,m -trivial (Def. 9.1.4). We will deduce this from Lemma 9.4.1. For that we need the following genG eralization of the technical Lemma 5.8.1 to the sheaf OZ,m : 9.6.2 Lemma. Let D ⊆ C be an open set such that Z ∩ D is discrete and closed in D, and let (Dn )n∈N be a sequence of bounded open sets such that (0) Z ∩ ∂Dn = ∅ for all n ∈ N; (1) Dn ⊆ Dn+1 for all n ∈ N;

9.6. Splitting over simply connected open sets (2)

n∈N

359

Dn = D

G (3) for each n, any function from OZ,m (Dn+1 ) can be approximated uniformly G on Dn by functions from OZ,m (D). G -cocycle over D such that, for each n ∈ N, the restriction Further let f be an OZ,m G G f |Dn is OZ,m -trivial (Def. 9.1.4). Then f itself is OZ,m -trivial .

Proof. We denote by · the norm of A and set dist(a, A \ G) = inf a − b b∈A\G

for a ∈ G.

Let U = {Uj }j∈I be the covering associated to f . By Proposition 9.1.6, after passing to a refinement, we may assume that each Uj is a relatively compact open disc in D and fjk ∈ OG Z.m (U j ∩ U k ) for all j, k ∈ I. Note that then, for each j ∈ I, there exists nj ∈ N with U j ⊆ Dn if n ≥ nj . (9.6.1) Moreover we may assume that for each compact set K ⊆ D there exists only a finite number of indices j ∈ I with Uj ∩ K = ∅.

(9.6.2)

To prove the lemma now it is sufficient to find a sequence fn n∈N of families fn = fn,j j∈I of functions fn,j ∈ OG Z,m (Dn+1 ∩ U j ) as well as a sequence (εn )n∈N of positive numbers, such that, for all n ∈ N, −1 fjk = fn,j fn,k

εn < max

z∈D n ∩U j

1 4

min

z∈D n ∩U j

on Dn+1 ∩ U j ∩ U k ,

j, k ∈ I,

(9.6.3)

dist fn,j (z), A \ G ,

j ∈ I,

(9.6.4)

fn,j (z) − fn−1,j (z) < εn−1 if n ≥ 1 ,

j ∈ I,

and

(9.6.5)

εn−1 if n ≥ 1. (9.6.6) 2 Indeed, then it follows from (9.6.1), (9.6.5) and (9.6.6) that, for all n, p ∈ N with nj ≤ n < p, εn
0, we may moreover assume that (9.6.6) holds for n = p + 1. From (9.6.7) we get −1 −1 −1 −1 fp+1,j fp+1,k = fp,j Ψ Ψfp,k = fp,j fp,k = fjk

362


on Dp+2 ∩ U j ∩ U k , i.e., (9.6.3) holds for n = p + 1. From (9.6.11) it follows that

−1 max fp+1,j − fp,j = max Ψfp,j fp,j − 1 fp,j < εp . D p ∩U j

D p ∩U j

Hence also (9.6.5) holds for n = p + 1.

Proof of Theorem 9.6.1. Since D is simply connected, by the Riemann mapping theorem we may assume that either D = C or D is the open unit disc. In both cases, there exists a sequence (Dn )n∈N of open discs Dn ⊆ C such that • Dn ⊆ D, • Dn ⊆ Dn+1 for all n ∈ N, • n∈N Dn = D, G • by the Runge approximation Theorem 9.5.3, for all n ∈ N, any f ∈ OZ,m (Dn+1 ) G (D). can be approximated uniformly on Dn by functions from OZ,m G Hence, by Lemma 9.6.2, it is sufficient to prove that for any OZ,m -cocycle f over G D, the restrictions f |Dn , n ∈ N, are OZ,m -trivial, which is indeed the case by Lemma 9.4.1.

Finally we point out the following special case of Theorem 9.6.1: 9.6.3 Corollary. Let D1 , D2 ⊆ C be two open sets such that D1 ∪ D2 is simply connected and Z ∩ (D1 ∪ D2 ) is discrete and closed in D1 ∪ D2 . Then, for each G G f ∈ OZ,m (D1 ∩ D2 ), there exist functions fj ∈ OZ,m (Dj ), j = 1, 2, such that −1 f = f1 f2 on D1 ∩ D2 .

9.7

Runge approximation. The general case

In this section A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, G1 A is the connected component of 1 in the group of invertible elements of A, and (Z, m) is a data of zeros (Def. 9.1.2). Here we generalize part (ii) of the Runge approximation Theorem 5.0.1 to G sections of OZ,m . For that we pass again to the Riemann sphere P1 = C ∪ {∞} and use the notation concerning the Riemann sphere introduced in the beginning of Section 5.10. Since, for each a ∈ P1 , Ta is biholomorphic between P1 \ {a} and the complex plane, from the Runge approximation Theorem 9.5.3 we immediately get: 9.7.1 Proposition. Let a ∈ P1 be a fixed point. Let D be a relatively compact open 1 subset P1 \{a} such that P1 \D is connected. Assume of with piecewise C -boundary 1 1 Z ∩ P \ {a} is discrete and closed in P \ {a} and Z ∩ ∂D = ∅.

9.7. Runge approximation. The general case

363

Let f ∈ OG Z,m (D) such that all values of f belong to the same connected component of G (which is always the case if D is connected). Then f can be approximated G uniformly on D by functions from OZ,m (P1 \ {a}). Moreover, from Corollary 5.9.3 we immediately get: 9.7.2 Proposition. Let a ∈ P1 and let D1 , D2 ⊆ P1 \ {a} be two open sets such that D1 ∪ D2 is simply connected and Z ∩ (D1 ∪ D2 ) is discrete and closed in D1 ∪ D1 . G G Then, for each f ∈ OZ,m (D1 ∩ D2 ), there exist functions fj ∈ OZ,m (Dj ), j = 1, 2, −1 such that f = f1 f2 on D1 ∩ D2 . 9.7.3 Lemma. Let D ⊆ P1 be an open set with piecewise C 1 -boundary and let U1 , . . . , Un be the connected components of P1 \ D. Let n ≥ 2 and let some points aj ∈ Uj , 1 ≤ j ≤ n, be fixed such that Z ∩ P1 \ {a1 , . . . , an } is discrete and closed in P1 \ {a1 , . . . , an }. Then, for each f ∈ OG there exist functions Z,m (D), 1 G 1 G fj ∈ OZ,m (P \ Uj ), 1 ≤ j ≤ n, and a function h ∈ OZ,m P \ {a1 , . . . , an } such that f = hfn . . . f1 on D. Proof. This proof is similar to the proof of Lemma 5.10.4. For 1 ≤ k ≤ n, we consider the following statement: 1 A(k): There exist functions fj ∈ OG ≤ j ≤ k, and a function Z,m P \ Uj , 1 G hk ∈ OZ,m D ∪ U1 \ {a1 } ∪ . . . ∪ Uk \ {ak } such that f = hk fk . . . f1 on D. Since

D ∪ U1 \ {a1 } ∪ . . . ∪ Un \ {an } = P1 \ {a1 , . . . , an },

then A(n) is the assertion of the lemma. Therefore it is sufficient to prove A(1) and the conclusions A(k) ⇒ A(k + 1), 1 ≤ k ≤ n − 1. Proof of A(1): Since 1 P \ U 1 ∪ D ∪ (U 1 \ {a1 }) = P1 \ {a1 } and

P1 \ U 1 ∩ D ∪ (U 1 \ {a1 }) = D, 1 G G P \ U 1 and h1 ∈ OZ,m D∪ from Proposition 9.7.2 we get functions f1 ∈ OZ,m U 1 \ {a1 } such that (9.7.1) f = h1 f1

on D. Since f is continuous and with values in G on D, since h1 is continuous and with values in G on D ∪ ∂U1 , since f1 is continuous and with values in G G on D \ ∂U1 and since (9.7.1) holds in D, it follows that f1 ∈ OZ,m (P1 \ U1 ), G h1 ∈ OZ,m D ∪ (U1 \ {a1 }) and (9.7.1) holds on D, i.e., assertion A(1) is valid.

364


Proof of A(k) ⇒ A(k +1): Let 1 ≤ k ≤ n−1 be given, assume that statement A(k) is valid, and let f1 , . . . , fk and hk be as in this statement. Since

P1 \ U k+1 ∪ D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } = P1 \ {ak+1 }

and

P1 \ U k+1 ∩ D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } = D ∪ U 1 \ {a1 } ∪ . . . ∪ U k \ {ak } ,

from Proposition 9.7.2 we get functions

G P1 \ U k+1 fk+1 ∈ OZ,m and

G D ∪ U 1 \ {a1 } ∪ . . . ∪ U k+1 \ {ak+1 } hk+1 ∈ OZ,m

such that hk = hk+1 fk+1 (9.7.2) in D ∪ U 1 \ {a1 } ∪ . . . ∪ U k \ {ak } . Since hk is continuous and with values in G on D, since hk+1 is continuous and with values in G on D ∪ ∂Uk+1 , since fk+1 is continuous and with values in G on D \ ∂Uk+1 and since (9.7.2) holds in D, it follows that

1 fk+1 ∈ OG Z,m P \ Uk+1 ,

hk+1 ∈ OG Z,m D ∪ U1 \ {a1 } ∪ . . . ∪ Uk+1 \ {ak+1 } and (9.7.2) holds on D. Since f = hk fk . . . f1 on D, this implies that f = hk+1 fk+1 fk . . . f1 on D, i.e., assertion A(k + 1) is valid.

9.7.4 Theorem. Let D ⊆ P1 be an open set with piecewise C 1 -boundary, and let U1 , . . . , Un be the connected components of P1 \ D. Let n ≥ 2 and let some points aj ∈ Uj , 1 ≤ j ≤ n, be fixed such that Z ∩ P1 \ {a1 , . . . , an } is discrete and closed in P1 \ {a1 , . . . , an } and Z ∩ ∂D = ∅ . Then the functions from OG be approximated uniformly on D by funcZ,m (D) can 1 tions from OZ,m P \ {a1 , . . . , an } , G .

9.8. The Oka-Grauert principle

365

Proof. This proof is similar to the proof of Theorem 5.10.5. If n = 1, the assertion of the theorem is that of Proposition 9.7.1. If n ≥ 2, then, by Lemma 9.7.3, each f ∈ OG Z,m (D) can be written in the form f = hfn . . . f1

on D,

(9.7.3)

1 1 G where fj ∈ OG Z,m (P \ Uj ), 1 ≤ j ≤ n, and h ∈ OZ,m P \ {a1 , . . . , an } . Let V be the interior of P1 \ Uj . Since the boundary of Uj is piecewise C 1 (as a part of the boundary of D), also the boundary of V is piecewise C 1 and V = P1 \ Uj . Since Uj is connected, P1 \ V = Uj is connected. Therefore, Proposition 9.7.1 can be applied to each Vj . Hence, each fj can be approximated uniformly 1on G G V = P1 \ Uj by functions from OZ,m P1 \ {aj }). Since OG P1 \ {aj } ⊆ OZ,m P \ 1 {p1 , . . . , pn } and D ⊆ P \ Uj ), this means in particular that each fj can be 1 G approximated uniformly on D by functions from OZ,m P \ {p1 , . . . , pn } . Since 1 G h belongs to OZ,m P \ {p1 , . . . , pn } and by (9.7.3), this implies the assertion of the theorem.

9.8

The Oka-Grauert principle

In this section A is a Banach algebra with unit 1, G is an open subgroup of the group of invertible elements of A, G1 A is the group of invertible elements of A, and (Z, m) is a data of zeros (Def. 9.1.2). Here we prove the following theorem: 9.8.1 Theorem. Let D ⊆ C be an open set such that Z ∩ D is discrete and closed G in D, and let f be an OZ,m -cocycle. Then: G -trivial. (i) If f is C G -trivial5 over D, then f is OZ,m G -trivial. (ii) If G is connected, then f is OZ,m

First we prove the following generalization of Lemma 5.11.3: 9.8.2 Lemma. Let D ⊆ C be an open set with piecewise C 1 -boundary such that Z ∩ D is finite and Z ∩ ∂D = ∅. Further, let U be a neighborhood of D and f an G OZ,m -cocycle over U . Suppose that at least one of the following two conditions is fulfilled: (i) f is C G -trivial over U . (ii) G is connected. G Then the restriction f |D is OZ,m -trivial. 5 This

G -trivial”. is not a misprint. We really mean “if f is C G -trivial” and not “if f is CZ,m The latter notion we even did not define, because we do not use it.

366


Proof. We may assume that D is connected. We proceed by induction over the number of connected components of C \ D. Beginning of induction: Suppose this number is 1, i.e., C \ D is connected. As the boundary of D is piecewise C 1 , then also C\D is connected, which means (cf., e.g., theorem 13.11 in Rudin’s book [Ru]) that D is simply connected. Therefore the assertion of the lemma follows from Theorem 9.6.1. (Even if none of the conditions (i) or (ii) is satisfied.) Hypothesis of induction: Assume, for some n ∈ N with n ≥ 2, the assertion of the lemma is already proved if the number of connected components of C \ D is n − 1. Step of induction: Assume that the number of connected components of C \ D is equal to n. Then, by Lemma 5.11.2, we can find a Cartan pair (D1 , D2 ) with D = D1 ∪ D2 satisfying conditions (1), (2), (3) (of this lemma). Since the boundaries of D1 , D2 , D1 ∩ D2 and D are piecewise C 1 , we can find a Cartan pair (D1 , D2 ) satisfying the same conditions (1), (2), (3) such that Dj ⊆ Dj and D1 ∪ D2 ⊆ U . G -trivial. Moreover, Then, again by Theorem 9.6.1, the cocycle f |D1 is OZ,m

since the number of connected components of C\D2 is equal to n−1, it follows from the hypothesis of induction that also f |D2 is OG -trivial. Hence both restrictions f |D1 and f |D2 are OG -trivial. By Proposition 9.1.9, this implies that f D ∪D is 1 2 G G OZ,m -equivalent to certain {D1 , D2 }, OZ,m -cocycle f . Since D1 ∩D2 ⊆ D1 ∩D2 , setting F = f12 |D1 ∩D2 , we get a function F ∈ OG Z,m (D 1 ∩ D 2 ). We claim that all values of F belong to the same connected component of G. If G = G1 A, this is trivial. If not, then condition (i) in the lemma under proof is satisfied, i.e., f is C G -trivial over U . As D1 ∪ D2 ⊆ U , then also the restriction f |D1 ∪D1 is C G -trivial over D1 ∪ D1 . Since f |D1 ∪D1 is C G -equivalent to f over G D1 ∪ D2 (it is even OZ,m -equivalent to f ), this implies that also f is C G -trivial, i.e., we can find Cj ∈ C G (Dj ), j = 1, 2, with = C1−1 C2 f12

on D1 ∩ D2 .

Hence condition (iii) in Lemma 5.11.1 is satisfied, and it follows (from this lemma) that all values of F belong to the same connected component of G. Since all values of F belong to the same connected component of G, it follows from the Cartan Lemma 9.3.7 that there exist functions Fj ∈ OG Z,m (D j ), j = 1, 2, with F = F1−1 F2 on Dj ∩ Dj . G -trivial. Since F |D1 ∩D2 = f12 |D1 ∩D2 , this means in particular that f D is OZ,m G Finally, as f D ∪D and f are OZ,m -equivalent and therefore f |D and f |D are 1

2

G G -equivalent, it follows that also f |D is OZ,m -trivial. OZ,m

9.9. Comments

367

Proof of Theorem 9.8.1. Take a sequence (Ωn )n∈N of bounded opensets with C 1 boundaries such that Ωn ⊆ Ωn+1 and Z ∩∂Ωn = ∅ for all n ∈ N and n∈N Ωn = D. Let Un be the union of all bounded connected components of C \ Ωn which are subsets of D (if there is any – otherwise Un := ∅), and set Dn = Ωn ∪ U n . Then also (Dn )n∈N is a sequence of bounded open sets with C 1 -boundaries such that Z ∩ ∂Dn = ∅, Dn ⊆ Dn+1 for all n ∈ N and n∈N Dn = D. Moreover this sequence has the important property that each bounded connected component of C \ Dn (if there is any) contains at least one point of C \ D. By the Runge approximation Theorem 9.7.4, this implies that, for each n, the functions from G OG Z,m (D n ) can be approximated uniformly on D n by functions from OZ,m (D). In particular we see that the sequence (Dn )n∈N has the properties (0)–(2) of Lemma G G 9.6.2. Therefore, by this lemma, f is OZ,m -trivial if each f |Dn is OZ,m -trivial. G That each f |Dn is OZ,m -trivial, follows from Lemma 9.8.2.

9.9

Comments

The results of this chapter are practically new, and here they are published for the first time, although they could be viewed as special cases of a much more general theory (see [FoRa] for finite dimensional groups, and [Le2, Le7] for infinite dimensional groups). However this is far not obvious. It is simpler to prove them again. Note also that some elements of the theory of cocycles with restrictions can be pointed out in the proofs in the papers [GR1, GR2]. The results of this chapter are used in the consequent chapters only.

Chapter 10

Generalized interpolation problems Here we prove further generalizations of the Weierstrass product theorem.

10.1

Weierstrass theorems

10.1.1 Theorem. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A, let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) G is connected. (ii) D is simply connected. Then there exist holomorphic functions hw : Uw → G, w ∈ Z, and a holomorphic function h : D \ Z → G such that hw fw = h

on Uw \ {w} .

(10.1.1)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions hw can be chosen so that, for each w ∈ Z, the functions hw − 1 and h−1 w − 1 have a zero of order ≥ mw at w. 10.1.2. In this theorem, it would be sufficient to claim that one of the functions hw − 1 or h−1 w − 1 has a zero of order ≥ mw at w. For the other one this follows automatically. Indeed, assume, for example, that this is the case for hw − 1. Then,

370

Chapter 10. Generalized interpolation problems

in a neighborhood of w, −1 h−1 w = 1 − (1 − hw ) =

∞

(1 − hw )j = 1 + (1 − hw )

j=0

∞

(1 − hw )j ,

j=0

which shows that also h−1 w − 1 has a zero of order ≥ mw at w. The topological conditions (i) and (ii) in Theorem 10.1.1 can be replaced by the more general condition that the problem can be solved continuously, i.e., there is the following Oka-Grauert principle: 10.1.3 Theorem. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A, let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Assume that: (iii) There exist continuous functions cw : Uw → G and a continuous function c : D \ Z → G such that cw fw = c on Uw \ {w}, w ∈ Z. Then there exist holomorphic functions hw : Uw → G, w ∈ Z, and a holomorphic function h : D \ Z → G such that hw fw = h

on Uw \ {w} .

(10.1.2)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions hw can be chosen so that, for each w ∈ Z, the functions hw − 1 and h−1 w − 1 have a zero of order ≥ mw at w. The first parts of theorems 10.1.1 and 10.1.3 coincide with theorems 5.13.1 and 5.13.2, respectively. The additional information at the end about the zeros of hw − 1 and (hw − 1)−1 has important consequences. For example, if the functions fw are meromorphic at w and, hence, h is meromorphic at each w (which is already clear by the first part of the theorem), then, for any given orders mw ∈ N∗ , the family of functions hw can be chosen so that h − fw has a zero of order mw , i.e., it can be achieved that arbitrarily prescribed finite pieces of the Laurent expansions of h and fw concide. The corresponding is true if the functions fw−1 are meromorphic. This is contained in the following immediate corollary of theorems 10.1.1 and 10.1.3: 10.1.4 Corollary. Assume, under the hypotheses of Theorem 10.1.1 or under the hypotheses of Theorem 10.1.3, two subsets Z+ and Z− of Z are given, where the cases Z+ = ∅, Z− = ∅ and Z+ ∩ Z− = ∅ are possible. Moreover assume that: • If w ∈ Z+ , then fw is meromorphic at w. • If w ∈ Z− , then fw−1 is meromorphic at w. Then, for any given family of positive integers mw , w ∈ Z, in the claims of these theorems, it can be shown that:

10.2. Right- and two-sided Weierstrass theorems

371

• If w ∈ Z+ , then f − fw has a zero of order mw at w. • If w ∈ Z− , then f −1 − fw−1 has a zero of order mw at w. Proof of Theorems 10.1.1 and 10.1.3. Let a family m = {mw }w∈Z of positive integers be given. Since Z is discrete and closed in D, then (Z, m) is a data of zeros in the sense of Definition 9.1.2. In terms of this definition, we have to find h ∈ OZ,m (D \ Z) and hw ∈ OZ,m (Uw ), w ∈ Z, with hw fw = h on Uw \ {w}. Choose neighborhoods Vw ⊆ Uw so small that Vw ∩ Vv = ∅ if w = v. It is sufficient to find h ∈ OZ,m (D \ Z) and hw ∈ OZ,m (Vw ), w ∈ Z, such that hw fw = h (10.1.3) on Vw \ {w}. Indeed, since Vw ∩ Z = Uw ∩ Z = {w}, then, by (10.1.3), each hw G admits an extension to a function from OZ,m (Uw ) such that (10.1.3) holds also for this extension. Set D1 = w∈Z Vw and D2 = D \ Z. Since the sets Vw are pairwise disjoint and Vw ∩ Z = {w}, the family of functions fw can be interpreted as a single holomorphic function f ∈ OG (D1 \ Z). Since Z ∩ D1 ∩ D2 = ∅, the restriction G f D1 ∩D2 belongs to OZ,m (D1 ∩ D2 ). Now, by Corollary 9.2.2, there exist h1 ∈ G G OZ,m (Dj ) and h2 ∈ OZ,m (D2 ) with f = h−1 1 h2 on D1 ∩ D2 . Setting hw = h1 Vw and h = h2 , we complete the proof. Since, in theorems 10.1.1 and 10.1.3, the multiplication by the functions hw is carried out from the left, we call these theorems left-sided Weierstrass theorems. There are also right- and two-sided versions.

10.2

Right- and two-sided Weierstrass theorems

If A is a Banach algebra with the multiplication “·”, then we can pass to the which consists of the same additive group A but with the Banach Algebra A multiplication “ · ” defined by a · b = b · a. In this way, from theorems 10.1.1 and 10.1.3 we get the following right-sided Weierstrass theorems: 10.2.1 Theorem. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A, let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) G is connected. (ii) D is simply connected.

372


Then there exist holomorphic functions hw : Uw → G, w ∈ Z, and a holomorphic function h : D \ Z → G such that fw hw = h

on Uw \ {w} .

(10.2.1)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions hw can be chosen so that, for each w ∈ Z, the functions hw − 1 and h−1 w − 1 have a zero of order ≥ mw at w. 10.2.2 Theorem. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A, let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function fw : Uw \ {w} → G. Moreover, we assume that: (iii) There exist continuous functions cw : Uw → G and a continuous function c : D \ Z → G such that fw cw = c on Uw \ {w}, w ∈ Z. Then there exist holomorphic functions hw : Uw → G, w ∈ Z, and a holomorphic function h : D \ Z → G such that fw hw = h

on Uw \ {w} .

(10.2.2)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions hw can be chosen so that, for each w ∈ Z, the functions hw − 1 and h−1 w − 1 have a zero of order ≥ mw at w. Finally, we present a two-sided Weierstrass theorem: 10.2.3 Theorem. Let A be a Banach algebra with unit 1, let G be an open subgroup of the group of invertible elements of A, let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and two holomorphic functions fw , gw : Uw \ {w} → G. Moreover, we assume that at least one of the following conditions is fulfilled: (i) G is connected. (ii) D is simply connected. Then there exist holomorphic functions hw : Uw → G, w ∈ Z, and a holomorphic function h : D \ Z → G such that fw hw gw = h

on Uw \ {w} .

(10.2.3)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions fw and gw can be chosen so that, for each w ∈ Z, the functions fw −1, fw−1 −1, gw −1 −1 − 1 have a zero of order ≥ mw at w. and gw

10.2. Right- and two-sided Weierstrass theorems

373

Proof. Let a family of positive integers mw , w ∈ Z, be given. Then from the leftsided Weierstrass Theorem 10.1.1 we get holomorphic functions hlw : Uw → G, w ∈ Z, and a holomorphic function hl : D \ Z → G such that hlw gw = hl

on Uw \ {w}

(10.2.4)

and the functions hlw − 1 have a zero of order ≥ mw at w. From the right-sided Weierstrass Theorem 10.2.1 we get holomorphic functions hrw : Uw → G, w ∈ Z, and a holomorphic function hr : D \ Z → G such that fw hrw = hr

on Uw \ {w}

(10.2.5)

and the functions hrw − 1 have a zero of order ≥ mw at w. Set h = hl hr and hw = hrw hlw , w ∈ Z. Then h ∈ OG (D \ Z), hw ∈ OG (Uw ) and fw hw gw = fw hrw hlw gw = hr hl = h

on Uw \ {w}

−1 − 1 have a zero of order ≥ mw and the functions fw − 1, fw−1 − 1, gw − 1 and gw at w.

10.2.4 Remark. Instead of conditions (i) or (ii) in Theorem 10.2.3 also the following condition would be sufficient (Oka-Grauert principle): (iii) There exist continuous functions cw : Uw → G, w ∈ Z, and a continuous function c : D \ Z → G such that fw cw gw = c on Uw \ {w}, w ∈ Z. But to prove this we would need a further generalization of the theory of multiplicative cocycles where the group G in Definition 9.1.2 is replaced by a fiber bundle of groups with characteristic fiber G. Also this generalization of Grauert’s theory is known in Complex analysis of several variables (see [FoRa] for finite dimensional groups, and [Le2, Le7] for infinite dimensional groups). But the direct proof of this result for the case of one complex variable would require a further chapter larger than Chapter 9. To keep the book shorter we omit this extension of the theory of cocycles. By the same arguments as in the case of Corollary 10.1.4, the above three theorems lead to the following two corollaries: 10.2.5 Corollary. Assume, under the hypotheses of one of the Theorems 10.2.1 or 10.2.2, two subsets Z+ and Z− of Z are given, where the cases Z+ = ∅, Z− = ∅ and Z+ ∩ Z− = ∅ are possible. Moreover assume that: • If w ∈ Z+ , then fw is meromorphic at w. • If w ∈ Z− , then fw−1 is meromorphic at w. Then, for any given family of positive integers mw , w ∈ Z, in the claims of these theorems, it can be shown that: • If w ∈ Z+ , then h − fw has a zero of order mw at w.

374


• If w ∈ Z− , then h−1 − fw−1 has a zero of order mw at w. 10.2.6 Corollary. Assume, under the hypotheses of Theorem 10.3.3, two subsets Z+ and Z− of Z are given, where the cases Z+ = ∅, Z− = ∅ and Z+ ∩ Z− = ∅ are possible. Moreover assume that: • If w ∈ Z+ , then the functions fw and gw are meromorphic at w. −1 • If w ∈ Z− , then the functions fw−1 and gw are meromorphic at w.

Then, for any given family of positive integers mw , w ∈ Z, in the claim of this theorem, it can be shown that: • If w ∈ Z+ , then the function h − fw gw has a zero of order mw at w. −1 −1 • If w ∈ Z− , then the function h−1 − gw fw has a zero of order mw at w.

10.3

Weierstrass theorems for G ∞ (E)- and G ω (E)-valued functions

Let E be a Banach space, let F ∞ (E) be the ideal in L(E) of operators which can be approximated by finite dimensional operators, and let F ω (E) be the ideal of compact operators in L(E). Throughout this section, ℵ stands for one of the symbols ∞ or ω, G ℵ (E) is the group of invertible operators in E which are of the form I + K, where K ∈ F ℵ (E). We first prove the following strengthening of the left-sided Weierstrass Theorem 5.14.1: 10.3.1 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function Aw : Uw \ {w} → G ℵ (E). Then there exist holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, and a holomorphic function H : D \ Z → G ℵ (E) such that Hw Aw = H

on Uw \ {w} .

(10.3.1)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions Hw can be chosen so that, for each w ∈ Z, the functions Hw − I and Hw−1 − I have a zero of order ≥ mw at w. Proof. Let a family of positive integers mw , w ∈ Z, be given. Since G ℵ (E) ⊆ GFIℵ (E), the functions Aw can be interpreted as functions with values in GFIℵ (E). Since the latter group is the group of invertible elements of a Banach algebra and since this group is connected, we can apply Theorem 10.1.1 to it and obtain w : Uw → GF ℵ (E), w ∈ Z, and a holomorphic function holomorphic functions H I ℵ H : D \ Z → GFI (E) such that w Aw = H H

on Uw \ {w}

(10.3.2)

10.3. G ∞ (E)- and G ω (E)-valued functions

375

w−1 − I have a zero of order ≥ mw w − I and H and , for each w ∈ Z, the functions H ℵ at w. If dim E < ∞ and therefore G (E) = GL(E) = GFIℵ (E), this completes the proof. Let dim E = ∞, and let λw : Uw → C, λ : D \ Z → C, Kw : Uw → F ℵ w = λw I + Kw and and K : D \ Z → F ℵ be the holomorphic functions with H ℵ H = λI + K. Then, passing to the factor algebra FI (E)/F (E) ∼ = C, we see: and H w are invertible, the functions λ and λw have no zeros. – Since H – It follows from (10.3.2) that λw = λ on Uw . w−1 − I have a zero of order ≥ mw at w, the functions w − I and H – Since H −1 λw − 1 and λw − 1 have a zero of order ≥ mw at w. w /λ and H := H/λ have the required properties. Hence the functions Hw := H Precisely in the same way, replacing the left-sided Theorem 10.1.1 by the right-sided Theorem 10.2.1, we get the corresponding left-sided result: 10.3.2 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function Aw : Uw \ {w} → G ℵ (E). Then there exist a holomorphic function H : D \ Z → G ℵ (E) and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z such that Aw H w = H

on Uw \ {w} .

(10.3.3)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions Hw can be chosen so that, for each w ∈ Z, the functions Hw − I and Hw−1 − I have a zero of order ≥ mw at w. Both theorems together again give a two-sided version: 10.3.3 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there are given a neighborhood Uw of w with Uw ∩ Z = {w} and two holomorphic functions Fw , Gw : Uw \ {w} → G ℵ (E). Then there exist a holomorphic function H : D \ Z → G ℵ (E) and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, such that Fw Hw Gw = H

on Uw \ {w} .

(10.3.4)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions Fw and Gw can be chosen so that, for each w ∈ Z, the functions Fw − 1, Fw−1 − 1, Gw − 1 and G−1 w − 1 have a zero of order ≥ mw at w. Proof. Let a family of positive integers mw , w ∈ Z, be given. Then from the leftsided Theorem 10.3.1 we get holomorphic functions Hwl : Uw → G ℵ , w ∈ Z, and a holomorphic function H l : D \ Z → G ℵ (E) such that Hwl Gw = H l

on Uw \ {w}

376


and the functions Hwl − 1 have a zero of order ≥ mw at w. From the right-sided Theorem 10.3.2 we get holomorphic functions Hwr : Uw → G ℵ (E), w ∈ Z, and a holomorphic function H r : D \ Z → G ∞ (E) such that Fw Hwr = H r

on Uw \ {w}

and the functions Hwr − 1 have a zero of order ≥ mw at w. Setting H = H l H r and Hw = Hwr Hwl , w ∈ Z, we get holomorphic functions H : D \ Z → G ℵ (E)), Hw : Uw → G ℵ (E) such that Fw Hw Gw = Fw Hwr Hwl Gw = H r H l = H

on Uw \ {w},

and the functions Fw − 1, Fw−1 − 1, Gw − 1 and G−1 w − 1 have a zero of order ≥ mw at w. By the same arguments as in the case of Corollary 10.1.4, the above three theorems lead to the following two corollaries: 10.3.4 Corollary. Assume, under the hypotheses of one of the theorems 10.3.1 or 10.3.2, two subsets Z+ and Z− of Z are given, where the cases Z+ = ∅, Z− = ∅ and Z+ ∩ Z− = ∅ are possible. Moreover assume that: • If w ∈ Z+ , then Aw is meromorphic at w. • If w ∈ Z− , then A−1 w is meromorphic at w. Then, for any given family of positive integers mw , w ∈ Z, in the claims of these theorems, it can be shown that: • If w ∈ Z+ , then H − Aw has a zero of order mw at w. • If w ∈ Z− , then H −1 − A−1 w has a zero of order mw at w. 10.3.5 Corollary. Assume, under the hypotheses of Theorem 10.3.3, two subsets Z+ and Z− of Z are given, where the cases Z+ = ∅, Z− = ∅ and Z+ ∩ Z− = ∅ are possible. Moreover assume that: • If w ∈ Z+ , then the functions Fw and Gw are meromorphic at w. • If w ∈ Z− , then the functions Fw−1 and G−1 w are meromorphic at w. Then, for any given family of positive integers mw , w ∈ Z, in the claim of this theorem, it can be shown that: • If w ∈ Z+ , then the function H − Fw Gw has a zero of order mw at w. −1 • If w ∈ Z− , then the function H −1 − G−1 w Fw has a zero of order mw at w.

10.3.6 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a neighborhood Uw of w with Uw ∩ Z = {w} and a holomorphic function Aw : Uw \ {w} → G ℵ (E) which is finite meromorphic at w. Then there exist a holomorphic function H : D \ Z → G ℵ (E),

10.4. Holomorphic G ∞ (E)-valued functions

377

which is finite meromorphic and Fredholm (Def. 4.1.1) at the points of Z, and holomorphic functions Hw : Uw → G ℵ (E), w ∈ Z, which are finite meromorphic and Fredholm at w, such that on Uw \ {w} .

Hw Aw = H

(10.3.5)

Moreover, for any given family of positive integers mw , w ∈ Z, the functions Hw can be chosen so that, for each w ∈ Z, the functions H − Aw and H −1 − A−1 w have a zero of order ≥ mw at w. Proof. By hypothesis, the functions Aw are finite meromorphic and Fredholm, and, by Corollary 4.1.3, also the functions A−1 w are finite meromorphic and Fredholm. In particular, these functions are meromorphic. Therefore, the assertion follows from Corollary 10.2.5 with Z+ = Z− = Z.

10.4

Holomorphic G ∞ (E)-valued functions with given principal parts of the inverse

In this section E is a Banach space, F ∞ (E) is the ideal in L(E) of the operators which can be approximated by finite dimensional operators, and G ∞ (E) is the group of invertible operators in E which are of the form I + K, where K ∈ F ∞ (E) (Def. 5.12.1). We first prove the following generalization of Corollary 4.3.3 to the Smith factorization theorem: 10.4.1 Lemma. Let w ∈ C, and let K : C \ L(E) be a rational function of the form K(z) =

−1

(z − w)n Kn ,

(10.4.1)

n=−m

where K−m , . . . , K−1 , 1 ≤ m < ∞, are finite dimensional operators. Then there exists a neighborhood U of w and a holomorphic operator function V : U → F ∞ (E) such that I + V (z) is invertible for all z ∈ U \ {w} and K is the principal part of the Laurent expansion of (I + V )−1 at w. Proof. Since the operators K−m , . . . , K−1 are finite dimensional, we can find a : C \ {w} → finite dimensional projector P in E such that K = P KP . Let K L(Im P ) be the function defined by K(z) = K(z)Im P for z ∈ C \ {w}. and, by Corollary 4.3.3, there exists a neighborhood U of w Then K = P KP : U → L(Im P ) such that A(z) and a holomorphic function A is invertible for all −1 at w. z ∈ U \ {w} and K is the principal part of the Laurent expansion of A Then A := I − P + P AP is the required function.

378


10.4.2 Theorem. Let D ⊆ C be an open set, and let Z be a discrete and closed subset of D. Suppose, for each w ∈ Z, there is given a rational function K : C → L(E) of the form −1 Kw (z) = (z − w)n Knw , (10.4.2) n=−mw

1 ≤ mw < ∞, are finite dimensional operators. Then there where exists a holomorphic function V : D → F ∞ such that I + V (z) is invertible for all z ∈ D \ Z and, for all w ∈ Z, the given function Kw is the principal part of the Laurent expansion of (I + V )−1 at w. w w , . . . , K−1 , K−m w

Proof. By Lemma 10.4.1, for each w ∈ Z, there exists a neighborhood Uw of w and a holomorphic operator function Vw : Uw → F ∞ (E) such that I + Vw (z) is invertible for all z ∈ Uw \ {w} and Kw is the principal part of the Laurent expansion of (I + Vw )−1 at w. By Corollary 10.2.5, now we can find holomorphic functions Hw : Uw → G ∞ (E), w ∈ Z, and a holomorphic function H : D \ Z → G ∞ (E), which is finite meromorphic at the points of Z, such that, for each w ∈ Z, on Uw \ {w},

Hw (I + Vw ) = H

and H −1 − (I + Vw )−1 has a zero of order mw at w. This implies, as K w is the principal part of the Laurent expansion of (I + Vw )−1 at w, that K w is also the principal part of the Laurent expansion of H −1 at w. It remains to set V = H − I. It is impossible to replace in Theorem 10.4.2 the prescribed function (10.4.2) by an arbitrary function of the form K(z) =

0

(z − w)n Kn ,

n=−m

where the operators K−m , . . . , K−1 are finite dimensional and K0 = 0. This follows from counterexample 4.3.4.

10.5

Comments

In such a generality the material of this chapter is published here for the first time. Less general versions of the interpolation theorems for matrix and operator functions were published earlier in [GR1, GR2].

Chapter 11

Holomorphic equivalence, linearization and diagonalization 11.1

Introductory remarks

11.1.1 Definition. Let X1 , X2 , Y1 , Y2 be Banach spaces such that X1 is isomorphic to X2 and Y1 is isomorphic to Y2 . Let D ⊆ C be an open set, and let S : D → L(X1 , Y1 ), T : D → L(X2 , Y2 ) be two holomorphic functions. The functions T and S are called holomorphically equivalent over D if there exist holomorphic functions E : D → L(X2 , X1 ) and F : D → L(Y1 , Y2 ) with invertible values such that T = F SE , on D . (11.1.1) We give an example. Let X be a Banach space and P (z) = z n I + z n−1 An−1 + . . . + A0 a polynomial where I is the identity operator in X and A0 , . . . , An−1 are arbitrary operators from L(X), n ∈ N∗ . Set X n := X ⊕ . . . ⊕ X . ,. + n times

Then the L(X n )-valued function ⎛ P (z) 0 ⎜ 0 I ⎜ P(z) := ⎜ . .. ⎝ .. . 0 0

··· ··· .. .

⎞ 0 0⎟ ⎟ .. ⎟ .⎠

···

I

380

Chapter 11. Holomorphic equivalence

is equivalent to the linear L(X n )-valued function zIX n − C where IX n is the identity operator in X n and ⎛ ⎞ 0 I 0 ··· 0 ⎜ 0 0 I ··· 0 ⎟ ⎟ ⎜ ⎜ .. . . .. ⎟ . . .. .. .. C := ⎜ . . ⎟ ⎟ ⎜ ⎝ 0 0 0 ··· I ⎠ −A0 −A1 −A2 · · · −An−1 Indeed, set

⎛

0 I .. .

··· ··· .. .

⎞ 0 0⎟ ⎟ .. ⎟ .⎠

z n−2 I

···

I

I zI .. .

⎜ ⎜ E(z) := ⎜ ⎝ z n−1 I

⎛ Bn−1 (z) Bn−2 (z) ⎜ −I 0 ⎜ ⎜ 0 −I F (z) := ⎜ ⎜ .. .. ⎝ . . 0 0

and

··· ··· ··· .. . ···

⎞ B1 (z) B0 (z) 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ .. .. ⎟ . . ⎠ −I 0

where B0 (z) := I Then

and Bk+1 (z) = zBk (z) + An−1−k P(z) = F (z) zIX n − C E(z)

for k = 0, 1, . . . , n − 2. , z ∈ D.

11.1.2 Definition. Let D ⊆ C be an open set, and X, Y, Z Banach spaces. Given an operator function T : D → L(X, Y ), we call the operator function T 0 : D −→ L(X ⊕ Z, Y ⊕ Z) 0 IZ the Z-extension of T . Here IZ is the identity operator of Z. According to the example given above a suitable extension of an operator polynomial is equivalent on C to a linear function. In the next section, a more general example is given for linearization of analytic operator functions by extension and equivalence.

11.2

Linearization by extension and equivalence

11.2.1 Theorem. Let D ⊆ C be a bounded open set with piecewise C 1 -boundary and such that 0 ∈ D. Let X be a Banach space, and T : D −→ L(X)

11.2. Linearization by extension and equivalence

381

a continuous operator function which is holomorphic in D. Denote by C(∂D, X) the Banach space of all X-valued continuous functions on ∂D endowed with the maximum norm. Let ⎧ ⎫ ⎨ ⎬ f (z) Z := f ∈ C(∂D, X) dz = 0 ⎩ ⎭ z ∂D

and define an operator A on C(∂D, X) by setting

1 I − T (ζ) f (ζ) dζ , (Af )(z) = zf (z) − 2πi

z ∈ ∂D.

(11.2.1)

∂D

Then X ⊕ Z and C(∂D, X) are isomorphic and the Z-extension of T is holomorphically equivalent on D to the linear operator function λ − A, λ ∈ D, i.e., there are holomorphic operator functions E : D → L X ⊕ Z, C(∂D, X) and F : D → L X ⊕ Z, C(∂D, X) with invertible values such that T (λ) 0 E(λ) λ − A F (λ) = , λ ∈ D. (11.2.2) 0 IZ Proof. Let τ : X → C(∂D, X) be the canonical embedding, i.e., (τ x)(z) = x for all x ∈ X and z ∈ ∂D. Furthermore, define an operator ω : C(∂D, X) → X by 1 f (z) ωf = dz . 2πi z ∂D

Since 0 ∈ D, then ωτ = IX and P := τ ω is the projector in C(∂D, X) with Im P = τ (X) and Ker P = Z. Let J : X ⊕ Z → C(∂D,X) be given by J(x, g) = τ x + g. Then J is invertible and J −1 f = ωf, (I − P )f . Next, consider on C(∂D, X) the following auxiliary operator (V f )(z) = zf (z) , Then D is in the resolvent set of V , where

f (z) (λ − V )−1 f (z) = , λ−z

z ∈ ∂D.

λ ∈ D, z ∈ ∂D.

(11.2.3)

Now we define functions E : D → L X ⊕ Z, C(∂D, X) and holomorphic operator F : D → L X ⊕ Z, C(∂D, X) , setting for λ ∈ D: E(λ) = (λ − V )−1 J and F (λ) = J −1 I − P B(λ)(I − P ) . It is clear that all values of E and F are invertible (as J is invertible, λI − V is invertible for λ ∈ D and P isa projector). Moreover we introduce a holomorphic operator function B : D → L C(∂D, X) defined by B(λ) = I + P V (λ − V )−1 − P V (λ − V )−1 T .

382


(Here T means the operator of multiplication by the operator function T .) Note that A = V − P V + P V T . Therefore

F (λ)(λ − A)E(λ) = F (λ) I + P V (λI − V )−1 − P V T (λI − V )−1 J . Since (λI − V )−1 is defined by multiplication by a scalar function, (λ − V )−1 commutes with the multiplication by T . Therefore it follows that

F (λ)(λI − A)E(λ) = F (λ)B(λ)J = J −1 I − P B(λ)(I − P ) B(λ) and further, since (I − P )B(λ) = I − P ,

F (λ)(λ − A)E(λ) = J −1 B(λ) − P B(λ)(I − P ) J

= J −1 I − P + B(λ)P J

= I − J −1 P J + J −1 B(λ)J J −1 P J . Since J −1 P J is the projector in X ⊕ Z with image X and kernel Z, it remains to prove that J −1 B(λ)Jx = T (λ)x for all x ∈ X and λ ∈ D. Let such x and λ be fixed. Then, by (11.2.3),

zx V (λ − V )−1 τ x (z) = , z ∈ ∂D, λ−z and

zT (z)x V (λ − V )−1 T x (z) = , z ∈ ∂D . λ−z By the Cauchy formula this implies that x 1 −1 ωV (λ − V ) τ x = dz = −x , 2πi λ−z

∂D

ωV (λ − V )−1 T x =

1 2πi

∂D

T (z)x dz = −T (λ)x , λ−z

and therefore P V (λ − V )−1 τ x = −τ x

and

P V (λ − V )−1 T x = −τ T (λ)x .

Hence

J −1 B(λ)Jx = J −1 B(λ)τ x = J −1 τ x + P V (λ − V )−1 τ x − P V (λ − V )−1 T x = J −1 τ T (λ)x = T (λ)x.


383

It can be shown (see [GKL]) that the spectrum σ(A) of the operator A defined by (11.2.1) is given by σ(A) = λ ∈ D T (λ) not invertible ∪ ∂D . The next theorem shows that for an operator function of the form λ − A the procedure of linearization by extension and equivalence does not simplify further the operator A and leads to operators that are similar to A. 11.2.2 Theorem. Let A1 and A2 be operators acting on the Banach spaces X1 and X2 , respectively, and suppose that for some Banach spaces Z1 and Z2 the extensions (λ − A1 ) ⊕ IZ1 and (λ − A2 ) ⊕ IZ2 are holomorphically equivalent over some open set D containing σ(A1 ) ∪ σ(A2 ) (here σ(Aj ) denotes the spectrum of Aj ). Then A1 and A2 are similar. More precisely: Let the holomorphic equivalence be given by λ − A1 0 0 λ − A2 = F (λ) E(λ) , λ ∈ D, (11.2.4) 0 IZ1 0 IZ2 let U be an open neighborhood of σ(A1 ) ∪ σ(A2 ) with piecewise C 1 -boundary such that U ⊆ D, and let πj : Xj ⊕ Zj → Xj , τj : Xj → Xj ⊕ Zj be the canonical projectors and embeddings, respectively. Then 1 S := (λ − A2 )−1 π2 F (λ)−1 τ1 dλ (11.2.5) 2πi ∂U

is a well-defined operator S : X1 → X2 (as ∂U is contained in the resolvent set of A2 ). Moreover, this operator is invertible, where 1 S −1 = π1 E(λ)−1 τ2 (λ − A2 )−1 dλ , (11.2.6) 2πi ∂U

and

SA1 S −1 = A2 .

Proof. From the equivalence (11.2.4) it follows that the integrands in (11.2.5) and (11.2.6) satisfy the following identities: (λ − A2 )−1 π2 F (λ)−1 τ1 = π2 E(λ)τ1 (λ − A1 )−1 ,

λ ∈ ∂U,

(11.2.7)

π1 E(λ)−1 τ2 (λ − A2 )−1 = (λ − A1 )−1 π1 F (λ)τ2 ,

λ ∈ ∂U.

(11.2.8)

Since, by Cauchy’s theorem, −1 π2 F (λ) τ1 dλ = π2 E(λ)−1 τ1 dλ = 0, ∂U

∂U

384


we get

1 A2 S = 2πi

(A2 − λ + λ)(λ − A2 )−1 π2 F (λ)−1 τ1 dλ

∂U

1 = 2πi

λ(λ − A2 )−1 π2 F (λ)−1 τ1 dλ

∂U

and therefore, by (11.2.7), A2 S =

1 2πi

λπ2 E(λ)τ1 (λ − A1 )−1 dλ

∂U 1 π2 E(λ)τ1 (λ − A1 + A1 )(λ − A1 )−1 dλ = 2πi ∂U 1 −1 = π2 E(λ)τ1 (λ − A1 ) dλ A1 . 2πi ∂U

Again by (11.2.7), this gives A2 S = SA1 . It remains to prove that S is invertible. To do this, we first note that, for λ ∈ D \ σ(A1 ) ∪ σ(A2 ) , we have the identities π1 E(λ)−1 τ2 (λ − A2 )−1 π2 F (λ)−1 τ1 − (λ − A1 )−1 = −π1 E(λ)−1

0 0

0

IZ2

F (λ)−1 τ1

(11.2.9)

and π2 E(λ)τ1 (λ − A1 )−1 π1 F (λ)τ2 − (λ − A2 )−1

0 = −π2 E(λ) 0 Indeed, from the equivalence (11.2.4) we get 0 (λ − A2 )−1 (λ − A1 )−1 = E(λ)−1 0 IZ1 0 and

(λ − A1 )−1 E(λ) 0

0 IZ1

F (λ) =

0

F (λ)τ2 . (11.2.10)

IZ1

0 IZ2

(λ − A2 )−1 0

F (λ)−1

0 IZ2

.

Multiplying the first equation from left by π1 and from the right by τ1 this gives (11.2.9). Multiplying the second equation from left by π2 and from the right by τ2 this gives (11.2.10).


385

Now we denote by T the operator defined by the right-hand side of (11.2.6), and we choose two neighborhoods U1 and U2 of σ(A1 ) ∪ σ(A2 ) with piecewise C 1 -boundary such that U 1 ⊆ U2 and U 2 ⊆ U . Then, by Cauchy’s theorem, S and T can be written also in the form 1 S= (λ − A2 )−1 π2 F (λ)−1 τ1 dλ 2πi ∂U1

and T =

1 2πi

π1 E(μ)−1 τ2 (μ − A2 )−1 dμ .

∂U2

By (11.2.7) and (11.2.8), this implies that 1 π2 E(λ)τ1 (λ − A1 )−1 dλ S= 2πi ∂U1

and T =

1 2πi

(μ − A1 )−1 π1 F (μ)τ2 dμ .

∂U2

Hence ST =

1 2πi

2

π2 E(λ)τ1 (λ − A1 )−1 (μ − A1 )−1 π1 F (μ)τ2 dμdλ.

∂U1 ∂U2

For λ ∈ ∂U1 and μ ∈ ∂U2 , we have the so-called resolvent equation (λ − A1 )−1 (μ − A1 )−1 =

(λ − A1 )−1 − (μ − A1 )−1 . μ−λ

(For the proof just multiply it from the left by λ − A1 and from the right by μ − A1 .) Therefore it follows that ST = A − B, where

A=

1 2πi

2

∂U1 ∂U2

and

B=

1 2πi

2

∂U1 ∂U2

π2 E(λ)τ1 (λ − A1 )−1 π1 F (μ)τ2 dμdλ μ−λ π2 E(λ)τ1 (μ − A1 )−1 π1 F (μ)τ2 dμdλ. μ−λ

386


We have A=

1 2πi

⎛ π2 E(λ)τ1 (λ − A1 )−1 ⎝

∂U1

1 = 2πi

1 2πi

∂U2

⎞ π1 F (μ)τ2 dμ⎠ dλ μ−λ

π2 E(λ)τ1 (λ − A1 )−1 π1 F (λ)τ2 dλ.

∂U1

Since the right-hand side of (11.2.10) is holomorphic in D and σ(A2 ) ⊆ U1 ⊆ U 1 ⊆ D, this implies 1 (λ − A2 )−1 π1 dλ = IX2 . A= 2πi ∂U1

Moreover, 1 B= 2πi

∂U2

⎛ ⎝ 1 2πi

∂U1

⎞ π2 E(λ)τ1 ⎠ dλ (μ − A1 )−1 π1 F (μ)τ2 dμ . μ−λ

Since ∂U2 ∩ U 1 = ∅, this implies that B = 0. We have now proved that ST = IX2 . In a similar way, using (11.2.9) instead of (11.2.10), one obtains T S = IX1 . For linear functions λ − A1 and λ − A2 global holomorphic equivalence on C means just that A1 and A2 are similar. This follows from the next corollary. 11.2.3 Corollary. Two operators A1 and A2 are similar if and only if some extensions of λ − A1 and λ − A2 are holomorphically equivalent on some open set containing σ(A1 ) and σ(A2 ). Proof. If A1 and A2 are similar, then, obviously, λ − A1 and λ − A2 are equivalent on C. Theorem 11.2.2 gives the reverse implication.

11.3

Local equivalence

11.3.1 Definition. Let X, Y be Banach spaces, let D ⊆ C be an open set, and let S, T : D → L(X, Y ) be meromorphic (Section 1.10.6). (i) Let w ∈ D. The functions T and S are called holomorphically equivalent at w if there exist a neighborhood Uw ⊆ D of w and holomorphic functions E : Uw → GL(X), F : Uw → GL(Y ) such that T = F SE

on Uw \ {w} .

(11.3.1)

(ii) The functions T and S are called locally holomorphically equivalent over D if they are holomorphically equivalent at each point of D.

11.3. Local equivalence

387

(iii) The functions T and S are called holomorphically equivalent over D if there exist holomorphic functions E : D → GL(X), F : D → GL(Y ) such that T (z) = F (z)S(z)E(z)

(11.3.2)

for all z ∈ D which are not singular for T and S. To point out the difference from local holomorphic equivalence over D, then we speak also about global holomorphic equivalence over D. If T and S have no singular points, part (iii) of this definition coincides with Definition 11.1.1 above. 11.3.2 Definition. (i) Let X be a Banach space, let D ⊆ C be an open set, and let f : D → B be meromorphic. Let w ∈ D. If f identically vanishes in a neighborhood of w, then we set ordw f = ∞. If not, then we denote by ordw f the uniquely determined integer such the Laurent expansion of f at w is of the form ∞ f (z) = fn (z − w)n with fordw f = 0 . n=ordw f

We call ordw f the order of f at w. (ii) Let X, Y be Banach spaces, let D ⊆ C be an open set, and let A : D → L(X, Y ) be meromorphic. Let w ∈ D. Then, for k ∈ Z, we denote by XA,w (k) the set of all x ∈ X such that there exists a neighborhood U of w and a holomorphic function f : U → X with f (w) = x

ordw (Af ) ≥ k.

and

Obviously, XA,w (k) is a linear subspace of X for all k ∈ Z. The family {XA,w (k)}k∈Z will be called the characteristic filtration of A at w. We set XA,w (∞) = XA,w (k) . k∈Z

Note that, obviously, XA,w (k + 1) ⊆ XA,w (k) and X=

for all k ∈ Z

XA,w (k) .

(11.3.3) (11.3.4)

k∈Z

11.3.3 Proposition. Let X, Y be Banach spaces, let D ⊆ C be an open set, let A, B : D → L(X, Y ) be meromorphic, and let w ∈ D. If A and B are holomorphically equivalent at w, then the characteristic filtrations of A and B at w are “isomorphic” in the following sense: There exists M ∈ GL(X) with XA,w (k) = M XB,w (k)

for all k ∈ Z .

(11.3.5)

388


Moreover, if U ⊆ D is a neighborhood of w and E : U → GL(Y ), F : U → GL(X) are holomorphic functions with EAF = B on U \ {w}, then (11.3.5) is valid for M = F (w). Proof. Let k ∈ Z∪{∞} and x ∈ XB,w (k) be given. By definition of XB,w (k), then, after shrinking U if necessary, we have a holomorphic function f : U → X with f (w) = x and ordw (Bf ) ≥ k. Set g = F f . Then g is a holomorphic function on U with g(w) = F (w)f (w) = F (w)x. Moreover, since E is holomorphic and invertible, it follows that ordw (Ag) = ordw (EAg) = ordw (EAF f ) = ordw (Bf ) ≥ k. Hence F (w)x ∈ XA,w (k). This proves “⊇” in (11.3.5). In the same way we prove that F −1 (w)XA,w (k) ⊆ XB,w (k)

for all k ∈ Z ,

i.e., “⊆”in (11.3.5).

11.3.4 Definition. Let X, Y be Banach spaces, let D ⊆ C be an open set, and let A : D → L(X, Y ) be meromorphic. (i) (For X = Y , this is Definition 4.1.1) Let w ∈ D. The function A is called finite meromorphic at w if the Laurent expansion of A at w is of the form A(z) =

∞

(z − w)n An ,

(11.3.6)

n=m

where (if m < 0) the operators Am , . . . , A−1 are finite dimensional. If, moreover, m ≤ 0 and A0 is a Fredholm operator, then A is called finite meromorphic and Fredholm at w. (ii) The function A is called a finite meromorphic Fredholm function on D if it is finite meromorphic and Fredholm at each point of D. (iii) If D is connected, and A is a finite meromorphic Fredholm function on D, then it follows from the stability properties of Fredholm operators (see, e.g., [GGK2]) that the index of A0 in (11.3.6) does not depend on w. We call it the index of A and denote it by ind A. Note that, obviously, this notion is invariant with respect to local holomorphic equivalence (Def. 11.3.1), i.e., if two meromorphic operator functions A and B are locally holomorphically equivalent, and A is a finite meromorphic Fredholm function, then B is a finite meromorphic Fredholm function. For finite meromorphic Fredholm functions the characteristic filtrations (Def. 11.3.2) are especially simple: 11.3.5 Proposition. Let X, Y be Banach spaces, let D ⊆ C be an open set, let A : D → L(X, Y ) be a finite meromorphic Fredholm function, and let w ∈ D. Then: (i) There exists m ∈ Z with XA,w (k) = X for k ≤ m.


389

(ii) For all k ≤ 0, the space XA,w (k) is closed and of finite codimension in X. (iii) dim XA,w (k) < ∞ for all k ≥ 1. Proof. We may assume that A = 0. Since A is finite meromorphic and Fredholm at w, then the Laurent expansion of A at w is of the form ∞

A(z) =

(z − w)n An ,

n=m

where m ≤ 0 is finite, A0 is a Fredholm operator, and (if m < 0) the operators Am , . . . , A−1 are finite dimensional. Then, it is clear that ordw (Af ) ≥ m for each X-valued holomorphic f in a neighborhood of w. Hence XA,w (k) = X if k ≤ m. This proves part (i). Since the operators Am , . . . , A−1 are finite dimensional (if m < 0), the space L := Ker Aν m≤ν≤−1

is closed and of finite codimension in X. Then, for each vector x ∈ L, the function fx := Ax admits a holomorphic extension fx to w, where fx (w) = A0 x. Hence L ⊆ XA,w (0). By (11.3.3) this proves (ii). Moreover, this implies that x ∈ XA,w (1) if x ∈ L and x ∈ Ker A0 . As A0 is a Fredholm operator, it follows that XA,w (1) ∩ L ⊆ Ker A0 and therefore dim XA,w (1) ≤ dim Ker A0 < ∞ .

By (11.3.3) this proves (iii).

11.3.6 Definition. Let X, Y be Banach spaces, let D ⊆ C be an open set, let A : D → L(X, Y ) be a finite meromorphic Fredholm function, let w ∈ D, and let A(z) =

∞

(z − w)n An

n=m

be the Laurent expansion of A at w. (i) By Proposition 11.3.5, α := XA,w (∞) < ∞ . The number α will be called the generic kernel dimension of A at w. (ii) As A0 is a Fredholm operator, we have the index ind A0 = dim Ker A0 − dim(Y / Im A0 ). The number β := α − ind A0 will be called the generic cokernel dimension of A at w.

390


(iii) If XA,w (k) = XA,w (0) for k ≤ 0, then we say that A has no negative powers. Assume this is not the case. Then, by Proposition 11.3.5, there are uniquely determined negative integers 0 > s1 > . . . > sp such that XA,w (0) ⊂ XA,w (s1 ) ⊂ XA,w (s2 ) ⊂ . . . ⊂ XA,w (sp ) = X ,

=

=

XA,w (ν) = XA,w (0) XA,w (ν) = XA,w (sj−1 )

=

=

if 0 > ν > s1 , if sj−1 ≥ ν > sj , 2 ≤ j ≤ p .

Again by Proposition 11.3.5 the dimensions 3

d1 := dim XA,w (s1 ) XA,w (0) , 3

dj := dim XA,w (sj ) XA,w (sj−1 ) ,

2≤j≤p

are finite. Set d = d1 + . . . + dp and denote by κ1 ≥ . . . ≥ κd the integers such that, for each 1 ≤ j ≤ p, precisely dj components of the vector (κ1 , . . . , κd ) are equal to sj . These numbers κ1 ≥ . . . ≥ κd will be called the negative powers of A at w. (iv) If XA,w (k) = XA,w (∞) for k ≥ 1, then we say that A has no positive powers. Assume this is not the case. Then, by Proposition 11.3.5, there are uniquely determined positive integers s1 > . . . > sp > 0 such that XA,w (∞) ⊂ XA,w (s1 ) ⊂ XA,w (s2 ) ⊂ . . . ⊂ XA,w (sp ) ⊂ XA,w (0) ,

=

=

XA,w (ν) = XA,w (∞) XA,w (ν) = XA,w (sj−1 ) XA,w (ν) = XA,w (sp )

=

=

=

if ν > s1 , if sj−1 ≥ ν > sj , 2 ≤ j ≤ p − 1 , if sp ≥ ν ≥ 1 .

Again by Proposition 11.3.5 the dimensions 3

d1 := dim XA,w (s1 ) XA,w (∞) , 3

dj := dim XA,w (sj ) XA,w (sj−1 ) ,

2 ≤ j ≤ p,

are finite. Set d = d1 +. . .+dp , and denote by κ1 ≥ . . . ≥ κd the integers such that, for each 1 ≤ j ≤ p, precisely dj components of the vector (κ1 , . . . , κd ) are equal to sj . These numbers κ1 ≥ . . . ≥ κd will be called the positive powers of A at w.


391

(v) Let α be the generic kernel dimension of A at w, and let β be the generic cokernel dimension of A at w. If A has no negative and no positive powers at w, then the pair (α, β) (defined in (i) and (ii)) will be called the numerical characteristic of A at w. If A has powers at w (negative or positive or both), and if κ1 > . . . > κn are all powers of A, then the vector (κ1 , . . . , κn , α, β) will be called the numerical characteristic of A at w. 11.3.7 Proposition. Let X, Y be Banach spaces, let D ⊆ C an open set, let A, A : D → L(X, Y ) be two finite meromorphic Fredholm functions, and let w ∈ D. Assume that A and A are holomorphically equivalent at w. Then A and A have the same numerical characteristic at w. Proof. Let

and

κ1 , . . . , κd , α, α

κ1 , . . . , κd , α , β

be the numerical characteristics of A and A , respectively. By definition, the numbers κ1 ≥ . . . ≥ κd and α depend only on the sequence 3

dim XA,w (k + 1) XA,w (k) , k ∈ Z. By Proposition 11.3.3 this sequence is invariant with respect to holomorphic equivalence at w. As A and A are holomorphically equivalent at w, this implies that α = α ,

d = d

and

κj = κj for 1 ≤ j ≤ d .

It remains to prove that β = β. Let A(z) =

∞

(z − w)n An

and

A (z) =

n=m

∞

(z − w)n An

n=m

be the Laurent expansions of A and A at w. By hypothesis, we have a neighborhood U ⊆ D of w and holomorphic functions E : U → GL(Y ), F : U → GL(X) such that A (z) = E(z)A(z)F (z) for all z ∈ U \ {w} . (11.3.7) From the stability properties of Fredholm operators it follows that ind A0 = ind A(z)

and

ind A0 = ind A (z)

for all z ∈ U \ {w} .

By (11.3.7) this implies that ind A0 = ind A0 . Hence β = α − ind A0 = α − ind A0 = β .

392


11.4

A theorem on local and global equivalence

In this section we prove that, under certain conditions, local holomorphical equivalence of meromorphic operator functions implies global holomorphical equivalence. We will state this result in a more general setting for Banach algebras. Throughout this section, A is a Banach algebra with unit 1, and GA is the group of invertible elements of A. 11.4.1 Definition. Let D ⊆ C be an open set, and let f : D → A be a meromorphic operator function (Section 1.10.6). Let Z be the set of all w ∈ D such that either f has a pole at w or f is holomorphic at w and f (w) ∈ GA. The function f will be called meromorphically invertible if Z is a discrete and closed subset of D, and f −1 (which is well defined and holomorphic on D \ Z) is meromorphic on D. The set Z then will be called the spectrum of f . If D ⊆ C is open and connected and A is the algebra of complex n × nmatrices, then it is clear that any meromorphic function f : D → A, which is invertible in at least one point, is meromorphically invertible. By Proposition 4.1.4 the same is true for all finite meromorphic Fredholm functions. 11.4.2 Theorem. Let D ⊆ C be an open set, let f, g : D → A be two meromorphically invertible meromorphic functions, and let G be an open subgroup of GA. Suppose f and g have the same spectrum Z, and the following condition (of local equivalence) is satisfied: For each w ∈ D there exist a neighborhood U ⊆ D and holomorphic functions aw , bw : U → G with aw f bw = g on U \ Z .

(11.4.1)

Then there exist holomorphic functions a, b : D → G with af b = g on D \ Z. Proof. By hypothesis we have an open covering {Uj }j∈I of D and families {aj }j∈I , {bj }j∈I of holomorphic functions aj , bj : Uj → G such that aj f bj = g Then

−1 f −1 a−1 i aj f = bi bj

on Uj \ Z ,

j ∈ I.

on Ui ∩ Uj \ Z ,

(11.4.2) i, j ∈ I.

(11.4.3)

Since both f and f −1 are meromorphic on D, for each w ∈ Z, we can choose nw ∈ N such that both functions (z − w)nw f (z)

and

(z − w)nw f −1 (z)

(11.4.4)

are holomorphic at w. Setting mw = 2nw + 1, w ∈ Z, we introduce a data of zeros g as well as the corresponding sheaf OZ,m (cf. Definition 9.1.2). Now, for each open set U ⊆ D, we denote by F(U ) the set of all holomorphic functions h ∈ OG (U ) such that the function f −1 hf,

11.4. Local and global equivalence

393

which is a well-defined G-valued holomorphic function on U \Z, admits a G-valued holomorphic extension to U . It is easy to see that in this way a OG -sheaf F over D is defined (Def. 9.1.1). We now prove that F is of finite order. It is clear that F(U \ Z) = OG (U \ Z) for each open set U ⊆ D. Since Z is a discrete and closed subset of D, therefore it is sufficient to prove that G OZ,m (U ) ⊆ F(U ) G (U ) be given. Since for all open sets U ⊆ D. Let an open set U ⊆ D and h ∈ OZ,m G mw = 2nw + 1, then, by definition of OZ,m , for each w ∈ Z, the function

h(z) − 1 (z − w)2nw +1

(11.4.5)

is holomorphic at w. Hence, for each w ∈ Z and z ∈ U \ Z, the function f −1 hf can be written in the form

f −1 (z)h(z)f (z) = 1 + f −1 (z) h(z) − 1 f (z) h(z) − 1 nw (z − w) = 1 + (z − w) (z − w)nw f −1 (z) f (z) . (z − w)2nw + 1 Since the functions (6.8.3) and (11.4.5) are holomorphic at w, for all w ∈ Z, this implies that h ∈ F(U ). Hence it is proved that F is of finite order. Therefore, Theorem 9.2.1 applies to F. As aj ∈ OG (Uj ), j ∈ I, the family −1 {ai aj }i,j∈I is an OG -trivial OG -cocycle. Since also bj ∈ OG (Uj ), j ∈ I, it follows G from (11.4.3), that {a−1 i aj }i,j∈I is even an F-cocycle over D. As it is O -trivial, −1 condition (i) in Theorem 9.2.1 is satisfied. It follows that {ai aj }i,j∈I is F-trivial, i.e., there are functions aj ∈ F(Uj ), j ∈ I with a−1 ai a−1 i aj = j

on Ui ∩ Uj ,

i, j ∈ I .

Therefore, setting a = aj aj

on Uj ,

j ∈I,

we get a global holomorphic function a : D → G. Moreover, we set b = f −1 a−1 g on D \ Z. Then, obviously, af b = af f −1 a−1 g = g and, on each Uj , we have

on D \ Z ,

−1 b = f −1 a−1 j aj g.

(11.4.6)

394


By (11.4.2), this implies that −1 a−1 b = f −1 j f bj

on Uj ,

j∈U.

(11.4.7)

−1 −1 As a−1 aj f extends to a G-valued holomorphic funcj ∈ F(Uj ), the function f −1 tion on Uj . Since also bj ∈ OG (Uj ), this implies together with (11.4.7) that b extends to a G-valued holomorphic function on D. In view of (11.4.6), this completes the proof.

11.5

The finite dimensional case

11.5.1 Proposition. Let D ⊆ C be a connected open set, let A be an n × m matrix of scalar meromorphic functions on D, and let P be the set of poles of A. Denote by rank A(z), z ∈ D \ P , the rank of A(z), and set r := max rank A(z) z∈D\P

Then

N := z ∈ D \ P rank A(z) < r

is a discrete and closed subset of D. Clearly, this follows from the Smith factorization Lemma 4.3.1, but it can be seen also more directly: Proof of Proposition 11.5.1. Take a point z0 ∈ D \ P with r = rank A(z0 ) . Then there is an r × r submatrix B of A such that det B(z0 ) = 0. Since D is connected and det B is meromorphic on D without poles in D \ P , then the set M := z ∈ D \ P det B(z) = 0 is discrete and closed in D. As N ⊆ M , it follows that N is discrete and closed in D. 11.5.2 Definition. We use the notation from the preceding Proposition 11.5.1, and we set Z = P ∪ N . Then the points in D \ Z will be called the generic points of A, and the number r will be called the generic rank of A. The points in Z will be called the non-generic points of A. Obviously, a quadratic matrix of scalar meromorphic functions has maximal generic rank if and only if it is meromorphically invertible in the sense of Definition 11.4.1. Therefore, the following lemma is a special case of Theorem 11.4.2:


395

11.5.3 Lemma. Let D ⊆ C be a connected open set, and let A, B be two r × rmatrices of scalar meromorphic functions of generic rank r on D which are locally holomorphically equivalent on D. Then A and B are globally holomorphically equivalent on D. Since any meromorphic matrix function is a finite meromorphic Fredholm function in the sense of Definition 11.3.4, we have the notion of the numerical characteristic (Def. 11.3.6) of a meromorphic matrix function. We now explain the relation of this to the Smith factorization Lemma 4.3.1. Let D ⊆ C be an open set, let A be an n × m matrix of scalar meromorphic functions on D, and let w ∈ D be a point such that A is not identically zero in a punctured neighborhood of w. Using the notion of equivalence (Def. 11.3.1), now the Smith factorization lemma can be stated as follows: There are uniquely determined integers κ1 ≥ . . . ≥ κr such that, at w, the matrix A is holomorphically equivalent to the (n × m)-matrix Δ 0 (11.5.1) 0 0 where Δ is the r × r diagonal matrix with the diagonal (z − w)κ1 , . . . , (z − w)κr . Obviously, r is the rank of the matrix (11.5.1) in a punctured neighborhood of w. Therefore r = rank A(z) (11.5.2) for all z in a punctured neighborhood of w (which again proves Proposition 11.5.2). Let p be the number of zero components of the vector (κ1 , . . . , κr ), and let ( κ1 , . . . , κ r−p ) be the vector obtained from (κ1 , . . . , κr ) omitting the zero components. Then it is easy to see that ( κ1 , . . . , κ r−p , m − r, n − r)

(11.5.3)

is the numerical characteristic (Def. 11.3.6) of the matrix (11.5.1) at w. Since the numerical characteristic is invariant with respect to holomorphic equivalence (Proposition 11.3.7), this implies that (11.5.3) is also the numerical characteristic of A at w. Conversely, this also shows that from the numerical characteristic of A at w one can obtain the powers (Def. 4.3.2) κ1 ≥ . . . ≥ κr in the Smith factorization theorem. We summarize: 11.5.4 Proposition. Let D ⊆ C be an open set, let A, B be two n × m matrices of scalar meromorphic functions on D and let w ∈ D be a point such that A and B are not identially zero in a punctured neighborhood of w. Then A and B have the same numerical characteristic at w if and only if they have the same powers at w.

396


11.5.5 Lemma. Let D ⊆ C be a connected open set, and let A be an n × m-matrix of scalar meromorphic functions on D. Let r be the generic rank of A, and assume that r > 0. Then there exist holomorphic matrix functions E : D → GL(n, C) and F : D → GL(m, C) such that EAF is a block matrix of the form B 0 EAF := , (11.5.4) 0 0 where B is an r × r-matrix of maximal generic rank of scalar meromorphic functions on D. Proof. By the Smith factorization Theorem 4.3.1, for each w ∈ D, we have a neighborhood Uw ⊆ D of w and holomorphic functions Ew : Uw → GL(n, C), Fw : Uw → GL(m, C) such that Δ(z) 0 , z ∈ Uw \ {w} , Ew (z)A(z)Fw (z) = 0 0 where Δ is the r × r diagonal matrix with the diagonal (z − w)κ1 , . . . , (z − w)κr . Setting R0 = z ∈ Cn zr+1 = . . . = zn = 0 and

K0 = z ∈ Cm z1 = . . . = zr = 0 ,

this implies that, for all w ∈ D and z ∈ Uw \ {w}, −1 Im A(z) = Ew (z)R0

and

Ker A(z) = Fw (z)K0 .

(11.5.5)

Let P be the set of poles of A and let N be the set of all z ∈ D \ P such that rank A(z) < r. If w ∈ D \ (P ∪ N ), then (11.5.5) holds also for z = w. Therefore, we have a uniquely determined holomorphic family R(z) z∈D of subspaces (see Def. 6.4.1) of Cn as well as a uniquely determined holomorphic family K(z) z∈D of subspaces of Cm such that R(z) = Im A(z)

and K(z) = Ker A(z)

for z ∈ D \ (P ∪ N ) .

Then, by Theorem 6.9.1, we can find holomorphic functions E : D → GL(n, C) and F : D → GL(m, C) such that E(z)R(z) = R0

and F (z)K0 = K(z)

for all z ∈ D .

Then, for z ∈ D \ (P ∪ N ),

Im E(z)A(z)F (z) = E(z)A(z)F (z)Cn = E(z)A(z)F (z) R0 ⊕ K0 = R0 and

Im E(z)A(z)F (z) = R0

and

Ker E(z)A(z)F (z) = K0 ,


397

i.e.,, over D \ (P ∪ N ), EAF is of the form (11.5.4). Since N is discrete and closed in D \ P , this implies by continuity that EAF is of the form (11.5.4) also in the points of N . 11.5.6 Lemma. Let D ⊆ C be a connected open set, and let A be an r × r-matrix of scalar meromorphic functions of generic rank r on D. Then there exist not identically vanishing scalar meromorphic functions ϕ1 , . . . , ϕr on D, such that the quotients ϕj /ϕj+1 , 1 ≤ j ≤ r − 1, are holomorphic on D, and A is locally holomorphically equivalent over D to the diagonal matrix with the diagonal ϕ1 , . . . , ϕr . Supplement: If w ∈ D and κ1 (w) ≥ . . . ≥ κr (w) is the vector of powers of A at w (Def. 4.3.2), then the functions ϕj (z) , (z − w)κj (w)

1 ≤ j ≤ r,

are holomorphic and = 0 in some neighborhood of w. Proof. Since the generic rank of A is maximal, by the Smith factorization Lemma 4.3.1, for each w ∈ D, there exist integers κ1 (w) ≥ . . . ≥ κr (w) such that A is holomorphically equivalent at w to the diagonal matrix function with the diagonal (z − w)κ1 (w) , . . . , (z − w)κr (w) .

(11.5.6)

Let Z be the set of points w ∈ D such that at least one of the numbers κ1 (w), . . . , κr (w) is different from zero. Since Z is discrete and closed in D, then by the Weierstraß product theorem we can find scalar meromorphic functions ϕ1 , . . . , ϕr on D which are holomorphic and = 0 on D \ Z and such that, for each w ∈ Z, there is a neighborhood Uw ⊆ D of w such that the functions hw j (z) :=

ϕj (z) , (z − w)κj (w)

1 ≤ j ≤ r,

are holomorphic and = 0 on Uw . Since κ1 (w) ≥ . . . ≥ κr (w), then the quotients ϕj /ϕj+1 , 1 ≤ j ≤ r − 1, are holomorphic on D. Denote by Δ the diagonal matrix with the diagonal ϕ1 , . . . , ϕk . Further, for w ∈ Z, we denote by Hw the diagonal matrix with the diagonal w hw 1 , . . . , hr .

Then each Hw is a holomorphic and invertible matrix function on Uw such that Hw−1 Δ is the diagonal matrix with the diagonal (11.5.6). Therefore A is holomorphically equivalent to Δ at each point w ∈ Z. Moreover, it is clear that A is holomorphically equivalent to Δ at each point w ∈ D \ Z, because, on D \ Z both A and Δ are holomorphic and of maximal rank r. Hence A and Δ are locally holomorphically equivalent over D.

398


Now we can prove the following global version of the Smith factorization lemma: 11.5.7 Theorem. Let D ⊆ C be a connected open set, and let A be an n × m matrix of scalar meromorphic functions on D. Let r be the generic rank of A, and assume that r > 0. Then there exist not identically vanishing scalar meromorphic functions ϕ1 , . . . , ϕr on D such that the quotients ϕj /ϕj+1 , 1 ≤ j ≤ r − 1, are holomorphic on D, as well as holomorphic matrix functions E : D → GL(n, C) and F : D → GL(m, C) such that Δ 0 EAF = (11.5.7) 0 0 where Δ is the r × r diagonal matrix with the diagonal ϕ1 , . . . , ϕr . Supplement: If w ∈ D and κ1 (w) ≥ . . . ≥ κr (w) is the vector of powers of A at w (Def. 4.3.2), then the functions ϕj (z) , (z − w)κj (w)

1 ≤ j ≤ r,

(11.5.8)

are holomorphic and = 0 in some neighborhood of w. Proof. By Lemma 11.5.5 we may assume that r = n = m. By Lemma 11.5.6, we can find not identically vanishing scalar meromorphic functions ϕ1 , . . . , ϕr on D, such that, for each w ∈ D, the functions (11.5.8) are holomorphic and = 0 in some neighborhood of w (and, hence, the quotients ϕj /ϕj+1 , 1 ≤ j ≤ r − 1, are holomorphic on D), and A is locally holomorphically equivalent over D to the diagonal matrix Δ with the diagonal ϕ1 , . . . , ϕr . Now the assertion follows from Lemma 11.5.3. Taking into account also Proposition 11.5 from this theorem we get the following corollary: 11.5.8 Corollary. Let D ⊆ C be a connected open set, and let A1 , A2 be two n × mmatrices of scalar meromorphic functions on D. Then the following are equivalent: (i) The matrix functions A1 and A2 are globally holomorphically equivalent over D. (ii) The matrix functions A1 and A2 are locally holomorphically equivalent over D. (iii) The matrix functions A1 and A2 have the same vectors of powers at each point in D (Def. 4.3.2). (iv) The matrix functions A1 and A2 have the same numerical characteristics at each point in D.


11.6

399

Local and global equivalence for finite meromorphic Fredholm functions

First we generalize the Smith factorization Theorem 4.3.1 to finite meromorphic Fredholm functions. For that we introduce the notion of a local diagonal power function: 11.6.1 Definition. Let X be a Banach space. By a projector in X we always mean a continuous linear projector in X, i.e., an operator P ∈ L(X) with P 2 = P . A family {Pj }j∈I of projectors in X will be called mutually disjoint if Pj Pk = 0 for all j, k ∈ I with j = k. Let X, Y be Banach spaces, and let w ∈ C. A function Δ : C\{w} → L(X, Y ) will be called a local diagonal power function at w if either Δ ∈ L(X, Y ) is a constant Fredholm operator or, for some n ∈ N∗ , Δ is of the form Δ(z) = Q0 B0 P0 +

n

(z − w)κj Qj Bj Pj ,

z ∈ C \ {w}

(11.6.1)

j=1

where κ1 ≥ . . . ≥ κn are integers = 0, P0 , . . . , Pn are non-zero mutually disjoint projectors in X, Q0 , . . . , Qn are non-zero mutually disjoint projectors in Y such that dim Ker P0 < ∞ and dim Ker Q0 < ∞ , (11.6.2) dim Im Pj = dim Im Qj = 1

if 1 ≤ j ≤ n ,

(11.6.3)

and Bj is an invertible operator from Im Pj onto Im Qj , 0 ≤ j ≤ n. 11.6.2 Remark. It is clear that any local diagonal power function Δ at w ∈ C is a finite meromorphic Fredholm function on C. Moreover it is easy to see that if Δ is written in the form (11.6.1), then

κ1 , . . . , κn , dim Ker P0 , dim Ker Q0

is the numerical characteristic of Δ at w (Def. 11.3.6). : C\ 11.6.3 Proposition. Let X, Y be Banach spaces, let w ∈ C, and let Δ, Δ {w} → L(X, Y ) be two local diagonal power functions at w. Then the following are equivalent: have the same numerical characteristic at w. (i) The functions Δ and Δ = EΔF . In (ii) There exist operators E ∈ GL(Y ) and F ∈ GL(X) with Δ particular, Δ and Δ are globally holomorphically equivalent over C in the sense of Definition 11.3.1 (iii).

400


Proof. The implication (ii)⇒(i) is a special case of Proposition 11.3.7. We prove (i)⇒(ii). Let Δ be written in the form (11.6.1), and let 0 P0 + 0 B Δ(z) =Q

n

j B j Pj , (z − w)κ j Q

z ∈ C \ {w}

j=1

Since Δ has the same numerical charbe the corresponding representation of Δ. acteristic at w, then n = n, κ j = κj , dim Ker P0 = dim Ker P0

and

0 = dim Ker Q0 . dim Ker Q

Then we can find G ∈ GL(Y ) and H ∈ GL(X) with Pj = H −1 Pj H

j = GQj G−1 and Q

for 0 ≤ j ≤ n,

and we obtain ⎛ 0 H −1 P0 + Δ(z) = G ⎝Q0 G−1 B

n

⎞ j H −1 Pj ⎠ H . (z − w)κj Qj G−1 B

j=1

j H −1 are invertible from Im Pj to Im Qj , further we can Since the operators G−1 B find operators Tj ∈ GL(Im Qj and Sj ∈ GL(Im Pj ) such that j H −1 Sj = Bj Tj G−1 B

for 0 ≤ j ≤ n .

Set T = (I − Q0 − . . . − Qn ) +

n

Qj Tj Qj

j=0

and S = (I − P0 − . . . − Pn ) +

n

Pj Sj Pj .

j=0

Then T ∈ GL(Y ), S ∈ GL(X) and ⎛ 0 H −1 P0 + T ⎝Q0 G−1 B

n

⎞ j H −1 Pj ⎠ S (z − w)κj Qj G−1 B

j=1

= Q0 B0 P0 +

n

(z − w)κj Qj Bj Pj = Δ(z) .

j=1

It remains to set E = GT and F = SH.


401

If X and Y are finite dimensional Banach spaces, then the Smith factorization Theorem 4.3.1 says that an L(X, Y )-valued meromorphic operator function, at any point, is holomorphically equivalent to a local diagonal power function. This can be generalized to arbitrary finite meromorhic Fredholm functions (Def. 11.3.4). 11.6.4 Theorem. Let X and Y be Banach spaces, let D ⊆ C be an open set and let A : D → L(X, Y ) be a finite meromorphic Fredholm function. Then, at each point w ∈ D, A is holomorphically equivalent to a local diagonal power function at w. Proof. Let w ∈ D be given, and let A(z) =

∞

(z − w)n An

n=m

be the Laurent expansion of A at w. Since the operators Am , . . . , A−1 are finite dimensional and A0 is a Fredholm operator, then there exists a projector PX in X such that dim (X/ Im PX ) < ∞ , if m ≤ j ≤ −1 , Aj PX = 0 and A0 Im P is an isomorphism between Im PX and A0 Im PX . Since A0 Im PX X is of finite codimension in Y , we can find a projector PY in Y with Im PY = A0 Im PX . Then there is a neighborhood U of w such that the function A : U \ {w} −→ L(Im PX , Y ) , defined by

A (z) = A(z)Im P , X

z ∈ U \ {w} ,

admits a holomorphic extension to w (which we also denote by A ) where A (w) = A0 . Let Im PX

A : U −→ L(Im PX , Im PY )

be the function defined by A = PY A . Then A (w) = A0 Im P is invertible X and, after shrinking U , we may assume that each A (z), z ∈ U , is invertible. Let QX := IX − PX and QY := IY − PY . Then, setting −1 PY , z∈U, E(z) = QY + A (z) A (w) we get a holomorphic function E : U → L(Y ) with E(w) = IY . After a further shrinking of U we may assume that E(z) ∈ GL(Y ) for all z ∈ U . Then E −1 A is holomorphically equivalent to A over U . Setting B(z) = E −1 (z)A (z)Im P , z∈U, X

we get a holomorphic function B : U −→ L(Im PX , Im PY ) ,

402


such that

B(z) = E −1 (z)A(z)Im P

Since

X

for z ∈ U \ {w} .

B(w) = A0 Im P

X

is an isomorphism between Im PX and Im PY , after a further shrinking of U , we may assume that B (z) is an isomorphism between Im PX and Im PY for all z ∈ U . Moreover, −1 E(z) Im PY = A (z) A (w) Im PY = Im A (z) , z ∈ U , and therefore Im PY = E −1 (z) Im A (z) = B(z) Im PX .

z∈U.

Hence E −1 (z)A(z) = PY B(z)PX + PY E −1 (z)A(z)QX + QY E −1 (z)A(z)QX for z ∈ U \ {w} and, setting F (z) = PX − PX (B(z))−1 PY E −1 (z)A(z)QX + QX , for all z ∈ U , we get a holomorphic function F : U → GL(X) such that E −1 (z)A(z)F (z) = PY B(z)PX + QY E −1 (z)A(z)QX ,

z∈U.

Since each B(z), z ∈ U , is invertible from Im PX to Im PY , and since the spaces Im QX and Im QY are finite dimensional, now the assertion of the lemma follows from the Smith factorization Theorem 4.3.1 applied to the operator function QY E −1 (z)A(z)Im Q . X

As a first consequence of Theorem 11.6.4 we obtain: 11.6.5 Corollary. Let X and Y be Banach spaces, let D ⊆ C be an open set, and let A, B : D → L(X, Y ) be two finite meromorphic Fredholm functions. Then, for each point w ∈ D, the following are equivalent: (i) The functions A and B have the same numerical characteristics at w. (ii) The functions A and B are holomorphically equivalent at w. Proof. That (ii) implies (i) is the statement of Proposition 11.3.7. To prove (i) ⇒ (ii), we consider a point w ∈ D and assume that A and B have the same numerical characteristics at w. By Theorem 11.6.4, then we have local diagonal power functions Δw,A and Δw,B at w which are holomorphically equivalent at w to A and B, respectively. By Proposition 11.3.7, then, at w, Δw,A has the same numerical characteristic as A, and Δw,B has the same numerical characteristic


403

as B. Since A and B have the same numerical characteristics at w, it follows that also Δw,A and Δw,B have the same numerical characteristics at w. Hence, by Proposition 11.6.3, the functions Δw,A and Δw,B are globally holomorphically equivalent over C. Since, at w ∈ D, the function A is holomorphically equivalent to Δw,A , and B is holomorphically equivalent to Δw,B , this implies that A and B are holomorphically equivalent at w. The following corollary is an immediate consequence of Theorem 11.6.41 : 11.6.6 Corollary. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, let A : D → L(X, Y ) be a finite meromorphic Fredholm function, and let P be the set of poles of A. Then there exist numbers n, m ∈ N and a discrete and closed subset Z of D with P ⊆ Z such that

3 dim Ker A(z) = n and dim Y Im A(z) = m if z ∈ D \ Z and dim Ker A(z) > n

and

#

dim Y Im A(z) < m

if z ∈ Z \ P .

In particular: If A(z0 ) is invertible for at least one point z0 ∈ D \ P , then there exists a discrete and closed subset Z of D with P ⊆ Z such that A(z) is invertible for all z ∈ D \ Z. 11.6.7 Definition. With the notation from the preceding corollary we define: The points in D \ Z will be called the generic points of A and the points in Z will be called the non-generic points of A. Also from Theorem 11.6.4 we get the following 11.6.8 Theorem. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, and let A : D → L(X, Y ) be a finite meromorphic Fredholm function. Let P be the set of poles of A, and let Z be the set of all non-generic points of A (Def. 11.6.7). Then there exist a holomorphic family {K(z)}z∈D of finite dimensional subspaces of X (Def. 6.4.1) and a holomorphic family {R(z)}z∈D of finite codimensional subspaces of Y such that Ker A(z) = K(z)

and

Im A(z) = R(z)

if z ∈ D \ Z ,

(11.6.4)

Ker A(z) ⊃ K(z)

and

Im A(z) ⊂ R(z)

if z ∈ Z \ P .

(11.6.5)

and

=

=

1 Note that this consequence can be obtained also in a more direct way without using the Smith factorization Theorem 4.3.1 which is contained in Theorem 11.6.4. Namely, by the same arguments as we deduced Theorem 11.6.4 from the Smith factorization lemma, one can deduce it from the simpler Proposition 11.5.1.

404


Proof. By Corollary 6.4.2 the families {Ker A(z)}z∈D\Z and {Im A(z)}z∈D\Z are holomorphic families of subspaces of X and Y , respectively. Therefore, over D \ Z, we can (and have to) define the required families K and R by (11.6.4). Now consider a point w ∈ Z. Then, by Theorem 11.6.4, there exist a neighborhood U ⊆ D of w, holomorphic functions E : U → GL(Y ), F : U → GL(X) and a local diagonal power function Δ(z) = Q0 B0 P0 +

n

(z − w)κj Qj Bj Pj ,

z ∈ C \ {w} ,

j=1

(with the properties as in Def. 11.6.1) such that A = EΔF

on U \ {w} .

By (11.6.4), then K(z) = F −1 (z) Ker P0 + . . . + Pn

and R(z) = E(z) Im Q0 + . . . + Qn

for z ∈ D \ Z. Therefore we can (and have to) define K(w) = F −1 (z) Ker P0 + . . . + Pn

and R(w) = E(w) Im Q0 + . . . + Qn .

Doing this for all points in Z, we obtain a holomorphic family {K(z)}z∈D of subspaces of X and a holomorphic family {R(z)}z∈D of subspaces of Y such that (11.6.4) is satisfied. Moreover, if w ∈ Z \ P , then κj > 0 for all 1 ≤ j ≤ n, which implies (11.6.5). 11.6.9 Definition. With the notation from the preceding theorem we define: The family {K(z)}z∈D will be called the smoothing of the kernel of A. The family {R(z)}z∈D will be called the smoothing of the image of A. With this definition, as an immediate consequence of Theorem 6.9.1 we obtain: 11.6.10 Theorem. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, let A : D → L(X, Y ) be a finite meromorphic Fredholm function, and let Z be the set of non-generic points of A. Let {K(z)}z∈D be the smoothing of the kernel of A, and let {R(z)}z∈D be the smoothing of the image of A, and let z0 ∈ D \ Z. Then there exist holomorphic functions E : D → GL(Y ) and F : D → GL(X) with K(z) = F (z)K(z0 )

and

R(z) = E(z)R(z0 )

for all z ∈ D.

(11.6.6)

11.6.11 Theorem. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, let A : D → L(X, Y ) be a finite meromorphic Fredholm function, and let Z be the set of non-generic points of A (Def. 11.6.7). Then there exist – a Banach space M ,


405

– a surjective operator Ψ ∈ L(X, M ) with dim Ker Ψ < ∞, – an injective operator Φ ∈ L(M, Y ) with finite codimensional and, hence2 , closed image in Y , – a finite meromorphic Fredholm function BM : D → L(M ) which is holomorphic and invertible on D \ Z, such that A is holomorphically equivalent to ΦBM Ψ over D. Proof. We use the notation of the preceding theorem; for z ∈ D \ Z we put A(z) = E −1 (z)A(z)F (z). By (11.6.6), then K(z0 ) = Ker A(z)

and R(z0 ) = Im A(z)

for z ∈ D \ Z.

(11.6.7)

Choose the required space M as a direct complement of K(z0 ) in X. Then, by the first relation in (11.6.6), F (z)M is a direct complement in E of K(z) = Ker A(z) for z ∈ D \ Z. Hence A(z)F (z)M = Im A(z) = R(z) for z ∈ D \ Z. By the second relation in (11.6.6), this implies that A(z)M = R(z0 )

for z ∈ D \ Z.

(11.6.8)

Moreover, since K(z0 ) is the kernel of A(z0 ) and R(z0 ) is the image of A(z0 ), A(z0 )M is an invertible operator from M onto R(z0 ). Set Φ = A(z0 )M . Then it follows from (11.6.8) that, by setting BM (z) = Φ−1 A(z) for z ∈ D \ Z, M is finite we obtain a holomorphic function BM : D \ Z → GL(M ). Since A meromorphic and Fredholm at the points of Z also as a function with values in L X, R(z0 ) , and since Φ−1 is an invertible operator from R(z0 ) to M , it follows that BM is finite meromorphic and Fredholm at the points of Z. Finally we choose Ψ as the projector from X onto M parallel to K(z0 ). Then it is clear from the definition of BM and (11.6.7) that = ΦBM (z)Ψ E −1 (z)A(z)F (z)M = A(z) for all z ∈ D \ Z. M M Moreover, by the first relation in (11.6.6), E −1 (z)A(z)F (z)K(z0 ) = E −1 (z)A(z)K(z) = {0} Hence

E −1 (z)A(z)F (z) = ΦBM (z)Ψ

for all z ∈ D \ Z .

for all z ∈ D \ Z ,

i.e., A and ΦBM (z)Ψ are holomorphically equivalent on D. 2 It

follows from the Banach open mapping theorem that Im Φ is closed if it is of finite codimension in Y .

406


11.6.12 Theorem. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, and let A, B : D → L(X, Y ) be finite meromorphic Fredholm functions. Then the following are equivalent: (i) The functions A and B have the same numerical characteristics at each point in D. (ii) The functions A and B are locally holomorphically equivalent on D. (iii) The functions A and B are globally holomorphically equivalent over D. Proof. The equivalence of (i) and (ii) follows from Corollary 11.6.5. The implication (iii) ⇒(ii) is trivial. It remains to prove (ii) ⇒ (iii). Assume the functions A and B are locally holomorphically equivalent on D. Then A and B have the same set of non-generic points. We denote this set by Z. By Theorem 11.6.12, we may assume that X = Y , and A(z), B(z) ∈ GL(X) for all z ∈ D\Z. Then, by Corollary 11.6.6, the functions A and B are meromorphically invertible in the sense of Definition 11.4.1. Therefore, it follows from Theorem 11.4.2 that they are globally holomorphically equivalent over D.

11.7

Global diagonalization of finite meromorphic Fredholm functions

Here we globalize the diagonalization Theorem 11.6.4. First we have to introduce an appropriate notion of a “global diagonal”. 11.7.1 Definition. Let M be a Banach space, and let D ⊆ C be a connected open set. A meromorphic function Δ : D → L(M ) will be called an invertible meromorphic diagonal function on D if, for some ω ∈ N ∪ {∞}, it is of the form Δ=I+

ω

(ϕj − 1)Pj

(11.7.1)

j=1

where: • {Pj }ω j=1 is a family of one-dimensional mutually disjoint projectors in M ; • {ϕj }ω j=r+1 is a family of not identically vanishing meromorphic functions on D such that the functions ϕj /ϕj+1 , r + 1 ≤ j ≤ ω − 1, are holomorphic on D; • if ω = ∞, then the following condition is satisfied (which ensures the conver-

11.7. Global diagonalization

407

gence of the infinite sum in (11.7.1)): for each compact set K ⊆ D, there exists ωK ∈ N such that the functions ϕj , j > ωK , are holomorphic on K, and ∞

Pj max ϕj (z) − 1 < ∞ . j=ωK +1

(11.7.2)

z∈K

11.7.2 Remark. If, in the preceding definition, ω < ∞ and if we set P := and Q = I − P , then Δ can be written in the form ⎞ ⎛ ω ϕj Pj ⎠ P , Δ=Q+P ⎝

ω j=1

Pj

j=1

which ∞ ”shows” the diagonal. For ω = ∞ this is impossible, because then the series j=1 ϕj Pj does not converge, at least not in the operator norm. In the sense of strong convergence however, this is sometimes possible. For example, let M be a separable Hilbert space with the scalar product (·, ·) and an orthonormal basis {ej }j∈N∗ , and let Pj (x) = (x, ej )ej for x ∈ H and j ∈ N∗ . Then, for each compact set K ⊆ D and sufficiently large ωK ∈ N, the series ∞ ϕj Pj x j=ωK +1

converges uniformly on K for each vector x ∈ H. This is even the case if instead of condition (11.7.2), we only require the following weaker condition: For each compact set K ⊆ D, there exists ωK ∈ N such that the functions ϕj , j > ωK , are holomorphic on K and sup j>ωK , z∈K

|ϕj (z)| < ∞ .

Therefore, then Δ can be written in the form Δ=

∞

ϕj Pj .

j=1

11.7.3 Remark. We use the notation of Definition 11.7.1. Then Δ is a finite meromorphic Fredholm function on D (Def. 4.1.1), which can be seen as follows: Let w ∈ D. Take a neighborhood Uw of w which is relatively compact in D. Then, by (11.7.2), we can find ωw ∈ N such that the functions ϕj , j > ωw , are

408


holomorphic in a neighborhood of U w , ∞

Set P =

ωw j=1

j=ωw

1

Pj max ϕj (z) − 1 < . 2 z∈U w +1

(11.7.3)

Pj and Q = I − P . Then Δ can be written in the form

Δ=Q I+

∞

(ϕj (z) − 1)Pj Q + P

j=ωw +1

ωw

ϕj (z)Pj P ,

j=1

where, by (11.7.3), Q I+

∞ j=ωw +1

(ϕj (z) − 1)Pj Q

Im Q

is holomorphic and invertible on Uw as a function with values in L(Im Q). This shows that Δ is a finite meromorphic Fredholm function on Uw . Moreover, the numerical characteristic of Δ at w can be found as follows: Let 1 ≤ j1 < . . . < jn ≤ ωw be the indices such that, for all 1 ≤ j ≤ ωw , = 0 if j ∈ {j1 , . . . , jn }, ordw ϕj =0 if j ∈ {j1 , . . . , jn }. Then

ordw ϕj1 , . . . , ordw ϕjn , 0 , 0

is the numerical characteristic of Δ at w. Now we construct invertible meromorphic diagonal functions with given numerical characteristics. 11.7.4 Lemma. Let M be an infinite dimensional Banach space, and let {Pj }∞ j=1 be an infinite sequence of mutually disjoint one-dimensional projectors in M . Let D ⊆ C be a connected open set, and let Z be a discrete and closed subset of D. w Suppose, for each w ∈ Z, a collection of non-zero integers κw 1 ≥ . . . ≥ κnw is ∗ given, nw ∈ N . Then there exists an invertible meromorphic diagonal function Δ : D → L(M ), where the projectors in (11.7.1) can be chosen from the family {Pj }∞ j=1 , such that • Δ is holomorphic and invertible on D \ Z; w • for w ∈ Z, κw 1 , . . . , κnw , 0, 0 is the numerical characteristic of Δ at w. ∗ Proof. Let {wν }ω ν=1 be the set Z numbered in some way, where ω ∈ N if Z is finite and ω = ∞ if Z is infinite. Set {1, 2, . . . , ω} if ω < ∞ , N∗ω = ∗ N if ω = ∞ .

11.7. Global diagonalization

409

By the Weierstrass product theorem (for example, by setting fw (z) = (z − w)κj in Theorem 2.7.1), we can find a sequence {φj }∞ j=1 of scalar meromorphic functions on D, which are holomorphic and = 0 on D \ {wν }ω ν=1 , and such that, for all ν ∈ N∗ω , ν for 1 ≤ j ≤ nwν , κw j (11.7.4) ordwν φj = 0 for all j ∈ N∗ with j > nν . wν ν As κw 1 ≥ . . . ≥ κnwν , then the quotients φj /φj+1 are holomorphic on D. First consider the case

m := sup nwν < ∞ . ν∈N∗ ω

Then Δ := I −

m j=1

Pj +

m

φj Pj = I +

j=1

m

(φj − 1)Pj

(11.7.5)

j=1

is a meromorphic diagonal function on D. Since each φj is holomorphic and = 0 on D \{wν }ω ν=1 , and by (11.7.4), we see that Δ has the same numerical characteristics as A. Now let sup nwν = ∞ . (11.7.6) ν∈N∗ ω

Then ω = ∞ and we have to modify the sequence {φj }∞ j=1 in order to obtain a sequence {ϕj }∞ of meromorphic functions on D satisfying also condition (11.7.2). j=1 Choose a sequence {Ks }∞ of compact subsets of D such that s=1 •

∞

Ks = D,

s=1

and, for each s ∈ N∗ , • Ks is contained in the interior of Ks+1 , • Ks is simply connected with respect to D (i.e., each connected component of C \ Ks contains at least one point of C \ D). Since the sequence {wν }∞ ν=1 is discrete and closed in D, each Ks contains at most a finite number of wν ’s. Therefore m(s) := sup nwν ν ∈ N∗ and wν ∈ Ks < ∞ for all s ∈ N∗ . Since the functions φj satisfy condition (11.7.4) and, on D\{wν }∞ ν=1 , they are holomorphic and = 0, it follows that φj is holomorphic and = 0 in a neighborhood of Ks if j > m(s), s ∈ N∗ . Set j ∈ N∗ . s(j) = max s ∈ N∗ j > m(s) ,

410


Then each φj is holomorphic and = 0 in a neighborhood of Ks(j) . Therefore, by the Runge approximation Theorem 5.0.1 for invertible functions, there exists holomorphic functions ψj : D → C \ {0} such that φj (z) 2−j (11.7.7) max − 1 < , j ∈ N∗ . z∈Ks(j) ψj (z)

Pj Set ϕj =

φj , ψj

j ∈ N∗ .

Since the functions φj satisfy condition (11.7.4) and, on D \ {wν }∞ ν=1 , they are holomorphic and = 0, then the same is true for the functions ϕj , i.e.,, for all ν ∈ N∗ , ν for 1 ≤ j ≤ nwν , κw j (11.7.8) ordwν ϕj = 0 for all j ∈ N∗ with j > nwν , and, on D \ {wν }∞ ν=1 , each ϕj is holomorphic and = 0. Since limj→∞ s(j) = ∞, each compact set K ⊆ D is contained in some Ks(j) . Therefore it follows from (11.7.7) that the sequence {ϕj }∞ j=1 satisfies condition (11.7.2). Therefore, setting Δ=I+

∞

(ϕj − 1)Pj ,

(11.7.9)

j=1

we can define a meromorphic diagonal function Δ on D. Since the functions ϕj satisfy condition (11.7.8), and, on D \ {wν }∞ ν=1 , they are holomorphic and = 0, we ∗ see that Δ is holomorphic and invertible on D \ {wν }∞ ν=1 and, at wν , ν ∈ N , Δ has the numerical characteristic (cf. Remark 11.7.3)

wν ν κw , . . . , κ , 0, 0 . nwν 1 11.7.5 Definition. Let X, Y be Banach spaces, and let D ⊆ C be a connected open set. A meromorphic function Δ : D → L(X, Y ) will be called a meromorphic diagonal function on D if there exists a Banach space M and an invertible meromorphic diagonal function ΔM : D → L(M ) such that Δ is of the form Δ = ΦΔM Ψ ,

(11.7.10)

where Ψ ∈ L(X, M ) is surjective with dim Ker Ψ < ∞, and Φ ∈ L(M, Y ) is injective with finite codimensional and, hence3 , closed image in Y , 3 It

follows from the Banach open mapping theorem that Im Φ is closed if it is of finite codimension in Y .

11.8. Comments

411

It is easy to see that each meromorphic diagonal function Δ on D (notation as in the preceding definition) is a finite meromorphic Fredholm function on D, where if (κ1 , . . . , κn , 0, 0) is the numerical characteristic of ΔM at some point w ∈ D (cf. Remark 11.7.3), then κ1 , . . . , κn , dim Ker Ψ, dim(Y / Im Φ)) is the numerical characteristic of Δ at w. 11.7.6 Theorem. Let X, Y be Banach spaces, let D ⊆ C be a connected open set, and let A : D → L(X, Y ) be a finite meromorphic Fredholm function. Then there exists a meromorphic diagonal function Δ : D → L(X, Y ) such that A and Δ are globally holomorphically equivalent over D. (Recall that then, by Theorem 11.6.12, the functions A and Δ have the same numerical characteristics.) Proof. Let Z be the set of non-generic points of A. Now we use the notation from Theorem 11.6.11. Then, by Lemma 11.7.4, there exists an invertible meromorphic diagonal function ΔM : D → L(M ) which has the same numerical characteristics as BM . By therorem 11.6.12, ΔM is globally holomorphically equivalent to BM over D, i.e., we have holomorphic functions T, S : D → GL(M ) with BM = T ΔM S. Let Φ(−1] be a left inverse of Φ, let Q be a projector from Y to Im Φ, let Ψ(−1) be a right inverse of Ψ, and let P be the projector from X to Im Ψ(−1) parallel to Ker Ψ. Define holomorphic functions F : D → GL(X) and D : D → Gl(Y ), setting F (z) = (I − P ) + P Ψ−1 S(z)ΨP

and E(z) = (I − Q) + QΦT (z)Φ(−1) Q

for z ∈ D. Then E(z)ΦΔM (z)ΨF (z) = QΦT (z)Φ(−1) QΦΔM (z)ΨP Ψ(−1) S(z)ΨP = ΦT (z)ΔM (z)S(z)Ψ = ΦBM (z)Ψ for all z ∈ D. Hence ΦBM Ψ and ΦΔM Ψ are holomorphically equivalent over D, where, by definition, ΦΔM Ψ is a meromorphic diagonal function. Since A and ΦBM Ψ are holomorphically equivalent over D, this completes the proof.

11.8

Comments

Holomorphic equivalence for polynomial matrix functions was introduced long ago (see [Ge], for holomorphic matrix functions see [BGR]). For operator functions it was considered probably for the first time in [Eni]. Developments concerned with this term can be found in [GKL, GS, Go3, GGK1] and the literature cited there. The first two sections together with the proofs are borrowed from [GGK1]. The local principle of section 8.3 is new and is published here for the first time. The global diagonalization theorems for finite meromorphic Fredholm functions were obtained in [Le4, Le5]. The corresponding local results were proved in [GS]. The fact that the kernel and the cokernel of such functions can be smoothed, which follows from [GS], was established already in [Go1].

Bibliography [A]

Arlt, D., Zusammenziehbarkeit der allgemeinen linearen Gruppe des Raumes c0 der Nullfolgen, Invent. Math. 1 (1966), 36–44.

[Ban]

Banach, S., Théorie des opérations linéaires, Chelsea Publishing Company, New York 1955.

[Bar]

Bari, N. K., Trigonometric series (Russian), Gosudarstvennoe Isdatelstvo Fisiko-Matematiˇceskoij Literaturoij, Moscou 1961.

[BGR]

Ball, J. A., Gohberg, I., Rodman, L., Interpolaion of rational matrix functions, Operator Theory: Advances and Applications 45, Birkh¨ auser, Basel-Boston, Mass., 1990, 605 pp.

[BGK]

Bart, H., Gohberg, I, Kaashoek, M. A., Minimal factorization of matrix and operator functions, Operator Theory: Advances and Applications 1, Birkh¨ auser, Basel-Boston, Mass., 1979, 228 pp.

[BGK1] Bart, H., Gohberg, I, Kaashoek, M. A., Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Functional Analyis, 68, No. 1 (1986), 1–42. [BGK2] Bart, H., Gohberg, I, Kaashoek, M. A., Invariants for Wiener-Hopf equvalence of analytic operator functions, Operator Theory: Advances and Applications 21, Birkh¨ auser, 1986, 317–355. [BGKR] Bart, H., Gohberg, I, Kaashoek, M. A., Ran, A.C.M. Factorization of matrix and operator functions: The state space method, Operator Theory: Advances and Applications 178, Birkh¨ auser, 2008. [Bu]

Bungart, L., On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers, Topology 7 (1967), 55–68.

[Ca]

Cartan, H., Sur les matrices holomorphes de n variables complexes, J. Math. Pures Appl. 19 (1940), 1–26.

[CG]

Clancey, K. F., Gohberg, I., Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications 3, Birkh¨ auser, Basel-Boston, Mass., 1981, 234 pp.

414

Bibliography

[CMM] Coifman, R. R., McIntosh, A., Meyer, Y., L’integrale de Cauchy definit un operateur borne sur L2 pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387. [Do1]

Douady, A., Un espace de Banach dont le group linéaire n’est pas connecxe, Indag. Math. 27 (1965), 787–789.

[Do2]

A. Douady, Le problème de modules pur sous-espaces analytiques compacts d’un espace analyique donné, Ann. Inst. Fourier, Grenoble 15 (1966), 1, 1–94.

[EMS1] Edelstein, I., Mitjagin, B., Semenov, E., The linear groups of C and L1 are contractible, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 18 (1970), 27–33. [EMS2] Edelstein, I., Mitjagin, B., Semenov, E., Letter to the editors, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 18 (1970), 213. [Eni]

Eni, V. M., Stability of the root-number of a bundle of operators (Rusˇ sian), Bul. Akad. Stiince RSS Moldoven, 1966 no. 4, 78–81.

[F]

Feldman, I. A., Wiener-Hopf operator equations and their application to the transport equation, Integral Equations and Operator Theory 3, 1 (1980), 43–61.

[Ge]

Gelfand, I. M., Lectures on linear algebra, with the collaboration of Z. Ya. Shapiro, translated from the second Russian edition by A. Shenitzer, reprint of the 1961 translation, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1989. vi+185 pp.

[FoRa]

Forster Ramspott, Okasche Paare von Garben nicht-abelscher Gruppen, Invent. math. 1 (1966), 260–286.

[Go1]

Gohberg, I. C, On linear operators depending analytically on a parameter (Russian), Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 629–632.

[Go2]

Gohberg, I. C., The factorization problem for operator-valued functions, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964) 1055–1082.

[Go3]

Gohberg, I. C., Certain questions of the spectral theory of finitely meromorphic operator functions (Russian), Izv. Akad. Nauk Armjan. SSR, Ser. Mat. 6 (1971), no. 2–3, 160–181; corrections, ibid. 7 (1972), no. 2, 152.

[Go4]

Goherg, I. C., A factorization problem in normed rings, functions of isometric and symmetric operators, and singular integral equations, Russian Mathematical Survey, London Mathematical Society, Pitman Press, 19, 1 ,1964, 63–115.

[GGK1] Gohberg, I., Goldberg, S., Kaashoek, M. A., Classes of linear operators. Vol. I, Operator Theory: Advances and Applications 49, Birkh¨ auser, Basel-Boston, Mass., 1990.

Bibliography

415

[GGK2] Gohberg, I., Goldberg, S., Kaashoek, M. A., Basic classes of linear operators, Birkh¨ auser, 2003. [GKL]

Gohberg, I. C., Kaashoek, M. A., Lay, D. C., Equivalence, linearization, and decomposition of holomorphic operator functions, J. Functional Analysis 28 (1978), no. 1, 102–144.

[GKS]

Gohberg, I., Kaashoek, M. A., Spitkovsky, I. M., An overview of matrix factorization theory and operator applications. Factorization and integrable systems (Faro, 2000), 1–102, Operator Theory: Advances and Applications 141, Birkh¨ auser, Basel-Boston, Mass., 2003.

[GKa]

Gohberg, I, Kaashoek, M.A., Constructive methods of Wiener-Hopf factorization, Operator Theory: Advances and Applications 21, Birkh¨ auser, 1986.

[GK]

Gohberg, I. C., Krein, M. G., Systems of integral equations on a half line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. (2) 14 (1960), 217–287.

[GKru] Gohberg, I, Krupnik, N., One-dimensional linear singular integral equations. Vol. I. Introduction, translated from the 1979 German translation by Bernd Luderer and Steffen Roch and revised by the authors, Operator Theory: Advances and Applications 53, Birkh¨ auser, Basel, 1992, 266 pp. [GLR]

Gohberg, I., Lancaster, P., Rodman, L., Matrix polynomials. Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

[GL1]

Gohberg, I. C., Leiterer, J, Holomorphic vector-valued functions of one variable. I. Functions on a compact set (Russian), Mat. Issled. 7 (1972), no. 4(26), 60–84, 252.

[GL2]

Gohberg, I. C., Leiterer, J., Holomorphic vector-valued functions of one variable. II. Functions in a domain (Russian), Mat. Issled. 8 (1973), no. 1(27), 37–58, 236.

[GL3]

Gohberg, I. C., Leiterer, J., Cocycles, operator-valued functions and families of subspaces (Russian), Mat. Issled. 8 (1973), no. 2(28), 23–56, 188.

[GL4]

Gohberg, I. C., Leiterer, J, The local principle for the factorization problem of continuous operator functions (Russian), Hommage au Professeur Miron Nicolescu pour son 70’me anniversaire, Rev. Roumaine Math. Pures Appl. 18 (1973), 1585–1600.

[GL5]

Gohberg, I. C., Leiterer, J., General theorems on the factorization of operator-valued functions with respect to a contour. I. Holomorphic functions (Russian), Acta Sci. Math. (Szeged) 34 (1973), 103–120, II. Generalizations (Russian), Acta Sci. Math. (Szeged) 35 (1973), 39–59.

416

Bibliography

[GM]

Gohberg, I. C., Markus, A. S., Two theorems on the gap between subspaces of a Banach space (Russian), Uspehi Mat. Nauk 14 (1959), no. 5 (89), 135–140.

[GR1]

Gohberg, I., Rodman, L., Analytic operator-valued functions with prescribed local data, Acta. Sci. Math. 45 (1983), 189–199.

[GR2]

Gohberg, I., Rodman, L., Analytic operator-valued functions with prescribed local data, Acta Sci. Math. (Szeged) 45 (1983), no. 1–4, 189–199.

[GS]

Gohberg, I. C., Sigal, E. I., An operator generalization of the logarithmic residue theorem and the theorem of Rouche, Math. USSSR Sbornik 13 (1971), 603–625.

[Gr1]

Grauert, H., Approximationss¨ atze f¨ ur holomorphe Funktionen mit Werten in komplexen R¨ aumen, Math. Ann. 133 (1957), 139–159.

[Gr2]

Grauert, H., Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133 (1957), 450–472.

[Gr3]

Grauert, H., Analytische Faserungen u ¨ber holomorph-vollst¨ andigen R¨ aumen, Math. Ann. 135 (1958), 263–273.

[HaGr]

Griffiths, P., Harris, J., Principles of Algebraic Geometry, Wiley and Sons, 1978.

[Ho]

L. H¨ ormander, An introduction to complex analysis in several variables, North-Holland 1990.

[KLH]

Spectral Methods in Linear Transport Theory, Operator Theory: Advances and Applications 5, Birkh¨ auser, Basel-Boston, Mass., 1982.

[K]

Krein, M. G., Integral equations on a half line with a kernel depending upon the difference of the arguments, Uspekhi Mat. Nauk 13, 5 (1958), 3–120 (Russian); English Trnsl., Amer. Math. Soc. Transl. (Series 2) 22 (1962), 163–288.

[KKM]

Krein, M., Krasnoselski, M., Milman, D., On the defect numbers of linear operators in Banach space and on some geometric problems (Russian), Sbornik Trud. Inst. Mat. Akad. Nauk Ukr. SSR 11 (1948), 97–112.

[Ku]

Kuiper, N. H., The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30.

[Le1]

¨ Leiterer, J., Uber Operatorfunktionen einer komplexen Ver¨ anderlichen, Dissertation B, Friedrich-Schiller-Universit¨ at Jena 1973.

[Le2]

Leiterer, J., Banach coherence and Oka’s principle, Contributions to the School on Global Analysis in Ludwigsfelde near Berlin, May 1976, Part 2, Akad. Wiss. DDR, Berlin 1977, VI.2–VI.67.

Bibliography

417

[Le3]

Leiterer, J., On factorization of matrix and operator functions (Russian), Soobˇcenija Akad. Nauk. Grus. SSR. 88 (1977), no. 3, 541–544.

[Le4]

Leiterer, J., Global diagonalization of meromorphic Fredholm operator functions, Akad. Wiss. DDR, Zentralinstitut f¨ ur Mathematik und Mechanik, Berlin, September 1977, 1–15.

[Le5]

Leiterer, J. Local and global equivalence of meromorphic operator functions. Part I, Math. Nachr. 83 (1978), 7–29,Part II, Math. Nachr. 84 (1978), 145–170.

[Le6]

Leiterer, J., Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91–109.

[Le7]

Leiterer, J., Oka’s principle for sheaves of infinite-dimensional nonabelian groups (Russian), in: Complex analysis and its applications (Russian), Nauka, Moscow 1978, 295–317.

[LiTz]

Lindenstrauss, J., Tzafiri, L., Classical Banach spaces. I and II, Classics in Mathematics, Springer 1996. Reprint of the 1977 and 1979 ed.

[Ma]

Markus, A. S., Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society.

[Mi]

Mitjagin, B. S., The homotopy structure of a linear group of a Banach space, Russian Math. Surveys 25 (1070), no. 5 (155), 59–103.

[MiE]

Mitjagin, B. S., Edelstein, I. S., The homotopy type of linear groups of two classes of Banach spaces, Funkcional. Anal. Appl. 4 (1970), 221–231.

[Mu]

Muschelishvili, N. I., Singular integral equations. Boundary problems of function theory and their application to mathematical physics, translated from the second (1946) Russian edition and with a preface by J. R. M. Radok, corrected reprint of the 1953 English translation, Dover Publications, Inc., New York, 1992. 447 pp.

[N1]

Neubauer, G., Der Homotopietyp der Automorphismengruppe in den R¨ aumen lp und c0 , Math. Ann. 174 (1967), 33–40.

[N2]

Neubauer, G., Review in Math. Rev.: MR0341523 (496274a), MR0341524 (496274b), MR0341525 (496274c), MR0341526 (496274d).

[Ok]

Oka, K., Sur les fonctions analytiques de plusieurs variables. II. Deuxième problème de Cousin, J. Sci. Hiroshima Univ., Ser. A9, 7–19 (1939).

[Pl]

Plemelj, J., Problems in the sense of Riemann and Klein, edited and translated by J. R. M. Radok, Interscience Tracts in Pure and Applied Mathematics, no. 16, Interscience Publishers John Wiley & Sons Inc., New York-London-Sydney 1964, vii+175 pp.

418

Bibliography

[Re]

Remmert, R., Funktionentheorie 2, Springer-Lehrbuch, 1992.

[Ro]

Rodman, L., An introduction to operator polynomials, Operator Theory: Advances and Applications 38, Birkh¨ auser, 1989.

[Ru]

Rudin, W., Real and complex analysis, McGraw-Hill 1987.

[Sh1]

Shubin, M. A., On the local principle in the factorization problem, Mat. Issled. 6 (1971), no. 1(28).

[Sh2]

Shubin, M. A., On holomorphic families of subspaces of a Banach space, translated from the Russian by K. Clancey, W. Kaballo and G. Ph. A. Thijsse, Integral Equations and Operator Theory 2 (1979), no. 3, 407– 420.

[Sz]

Szeg¨ o, G., Orthogonal Polynomials, American Math. Soc., Providence, 1975.

[ViMa]

Virozub, A.I., Macaev, V.I., A theorem on the canonical factorization of operator functions (Russian), Matem. Issled. Kishinev 8 (1973), no. 3, 145–150.

Index C 1 -contour, 8 C 1 -parametrization, 8 C G -equivalence of cocycles, 135 C k -contour closed connected, 72 C k -parametrization, 72 F-cocycle, 344 trivial, 344 F-equivalence, 344 G ω (E) and G ∞ (E), 151 OG -equivalence of cocycles, 135 OG -sheaf, 343 Abel’s lemma, 19 approximation Mergelyan, 38 Runge, 38 Runge for G-valued functions, 147 Runge for invertible scalar functions, 50 block matrix, 297 block T¨ oplitz operator, 297 block T¨ oplitz operator defined by a continuous operator function, 298 canonical factorization w.r.t. a contour, 220 functions with values in a Banach algebra, 221 canonical factorization w.r.t. the real line, 324 Cartan lemma, 129, 141

Cartan pair, 129 Cauchy formula, 12 Cauchy formula for continuous functions, 31 Cauchy integral, 61 Cauchy integral theorem, 11 homotopy version, 11 Cauchy-Riemann derivative, 29, 30 Cauchy-Riemann equation, 17 class C α , 33 class C k+α on D± , 73 class C k on Γ, 73 class C k+α on Γ, 73 closed C k -contour, 72 closed contour, 8, 9 cocycle (U, C G )-cocycle, 131 (U, OG )-cocycle, 131 C G -cocycle, 135 F-cocycle, 344 F-trivial, 344 OE -cocycle, 43 E OZ,m -cocycle, 52 OG -cocycle, 135 ∞ OG (E) -cocycle, 152 ∞ OG (E) -trivial, 152 ω OG (E) -cocycle, 152 ω OG (E) -trivial, 152 E O -cocycle, 123 additive, 43 induced on a refinement, 134, 344 multiplicative, 131 trivial, 131

420 with values in a family of subspaces, 186 cocycle condition, 131 complexly differentiable, 1 complexly differentiable function, 1 continuous equivalence of cocycles, 135 contour C 1 -contour, 8 closed, 9 closed C k -contour, 72 closed connected C k -contour, 72 piecewise C 1 -contour, 8 convolution, 306 between L1 R, L(H) and L2 (R, H), 315 of functions with values in a Banach algebra, 306 Cousin problem additive, 43 multiplicative, 132 data of zeros, 52 derivative of f w.r.t. Γ, 73 diagonal function local power, 399 meromorphic, 410 meromorphic invertible, 406 equivalence of F-cocycles, 344 of cocycles, 131, 135 of holomorphic operator functions, 379 of meromorphic operator functions at a point, 386 of meromorphic operator functions, global, 387 of meromorphic operator functions, local, 386 w.r.t. a contour, 221 essential singularity of f , 26 extension of operator functions, 380

Index factorization w.r.t. a contour canonical, 220 global, 84 local, 93, 220 of operator functions, 220 of scalar functions, 84 factorization w.r.t. the real line, 324 canonical, 324 family of subspaces complemented, 206 continuous, 172 holomorphic, 185 injective, 203 Shubin family, 204 filtration characteristic of a meromorphic operator function at a point, 387 of an operator function relative to a closed contour, 255 finite meromorphic, 97, 388 finite meromorphic and Fredholm, 97, 388 finite meromorphic Fredholm function, 388 function complexly differentiable, 1 Hölder continuous with exponent α, 33 Hölder-α continuous, 33 holomorphic, 1 of class C α , 33 Wiener, 60 gap metric, 159 generalized inverse, 212 generalized invertible, 212 generic cokernel dimension of a finite meromorphic Fredholm function, 389 generic kernel dimension of a finite meromorphic Fredholm function, 389

Index generic point of a finite meromorphic Fredholm function, 403 of a meromorphic matrix function, 394 generic rank of a meromorphic matrix function, 394 H¨ older continuous, 33 H¨ older continuous with exponent α, 33 H¨ older-α continuous, 33 Hahn-Banach criterion, 15 holomorphic, 1 holomorphic equivalence of cocycles, 135 holomorphic function, 1 holomorphic function with isolated singularities, 28 homotopy version of the Cauchy integral theorem, 11 index of a finite meromorphic Fredholm function, 388 of a function which is finite meromorphic and Fredholm at a point, 97 of a function w.r.t. a contour, 86 w.r.t. a contour of a function which is finite meromorphic Fredholm, 105 Isolated singularities, 25 isolated singularity, 25 kernel function of a symbol, 326 of a Wiener-Hopf operator, 326 Laurent series, 23 length of a contour, 10 Liouville’s theorem, 2 local diagonal power function, 399 local factorization w.r.t. a contour, 220

421 of scalar functions, 93 local principle additive, 82 multiplicative, 94 local splitting, additive, 82 maximum pronciple, 3 Mergelyan approximation, 38 meromorphic, 28 meromorphically invertible, 392 Mittag-Leffler theorem, 44, 53, 56 multiplicity of a partial index of an operator function, relative to R, 325 of a partial index of an operator function, relative to a closed contour, 220, 255 numerical characteristic of a meromorphic Fredholm function at a point, 389 order of the pole z0 , 26 order of the zero z0 , 26 orientation defined by, 11 orientation of a contour, 9 oriented as the boundary of, 11 oriented by, 11 parametrization C 1 -parametrization, 8 C k -parametrization, 72 piecewise C 1 -parametrization, 9 partial index non-zero of an operator function, relative to a closed contour, 220 of an operator function relative to a closed contour, 254 of an operator function, relative to R, 325 piecewise C 1 -boundary, 11 piecewise C 1 -contour, 8 piecewise C 1 -parametrization, 9

422 pole, 26 Pompeiju operator, 33 power series absolutely convergent, 19 convergent, 19 powers of a meromorphic Fredholm function at a point, 389 rational function with values in a Banach space, 223 regular point, 28 regularity of ∂, 37 removable singularity of f , 25 residuum, 25 resolution of a family of subspaces, 185 globally short, 192 injective, 192 Runge approximation for G-valued functions, 147 for invertible scalar functions, 50 for scalar functions, 38 section (Γ, κ)-section of an operator function defined on a closed contour Γ, 251 (Γ, κ)-section of an operator function defined on a closed contour through (z, v), 258 of a family of subspaces, continuous, 172 of a family of subspaces, holomorphic, 184 of a sheaf, 343 sheaf OG -sheaf, 343 defined by a data of zeros, 344 of finite order, 344 simply connected, 11 singular point, 28 smoothing of the image of a finite meromorphic Fredholm function, 404

Index smoothing of the kernel of a finite meromorphic Fredholm function, 404 splitting of holomorphic cocycles, 132 splitting R-spaces and algebras, 224 splitting of multiplicative cocycles, 131 splitting w.r.t. a contour, 60 global, 60 over U , 82 vanishing at infinity, 60 strong maximum principle, 3 symbol of a Wiener-Hopf operator, 326 of a kernel function, 326 Töplitz operator, 297 trivial cocycle, 131, 152 triviality of F-cocycles, 344 uniqueness of holomorphic functions, 2 Weierstrass product theorem, 51 generalization, 54–56, 153–157, 369–376 Wiener functions, 60, 222 Wiener-Hopf operator defined by an operator function on a closed contour, 270, 287 defined by an operator function on the real line, 326 of a kernel function, 326 of a symbol, 326 zero of, 25

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